Additional Questions - Chapter 2 Kinematics 11th Science Guide Samacheer Kalvi Solutions - SaraNextGen [2024-2025]

Updated On May 15, 2024

By SaraNextGen

Additional Questions Solved
Multiple Choice Questions
Question 1.
The radius of the earth was measured by -
(a) Newton
(b) Eratosthenes
(c) Galileo
(d) Ptolemy
Answer:
(b) Eratosthenes
Question 2
The branch of mechanics which deals with the motion of objects without taking force into account is -
(a) kinetics
(b) dynamics

(c) kinematics
(d) statics
Answer:
(c) kinematics
Question 3.
If the coordinate axes $(\mathrm{x}, \mathrm{y}, \mathrm{z})$ are drawn in anticlockwise direction then the co-ordinate system is known as -
(a) Cartesian coordinate system
(b) right handed coordinate system
(c) left handed coordinate system
(d) cylindrical coordinate system
Answer:
(b) right handed coordinate system
Question 4.
The dimension of point mass is -
(a) 0
(b) 1
(c) 2
(d) $\mathrm{kg}$
Answer:
(a) 0
Question 5.
If an object is moving in a straight line then the motion is known as -
(a) linear motion
(b) circular motion
(c) curvilinear motion
(d) rotational motion

Answer:
(a) linear motion
Question 6.
An athlete running on a straight track is an example for the whirling motion of a stone attached to'a string is a -
(a) linear motion
(b) circular motion
(c) curvilinear motion
(d) rotational motion
Answer:
(a) linear motion
Question 7.
The whirling motion of a stone attached to a string is a -
(a) linear motion
(b) circular motion
(c) curvilinear motion
(d) rotational motion

Answer:
(b) circular motion
Question 8.
Spinning of the earth about its own axis is known as -
(a) linear motion
(b) circular motion
(c) curvilinear motion
(d) rotational motion
Answer:
(d) rotational motion
Question 9.
If an object executes a to and fro motion about a fixed point, is an example for -
(a) rotational motion
(b) vibratory motion
(c) circular motion
(d) curvilinear motion
Answer:
(b) vibratory motion
Question 10.
Vibratory motion is also known as -
(a) circular motion
(b) rotational motion
(c) oscillatory motion
(d) spinning
Answer:
(c) oscillatory motion

Question 11.
The motion of satellite around the earth is an example for -
(a) circular motion
(b) rotational motion
(c) elliptical motion
(d) spinning
Answer:
(a) circular motion
Question 12.
An object falling freely under gravity close to earth is -
(a) one dimensional
(b) circular motion
(c) rotational motion
(d) spinning motion
Answer:
(a) one dimensional

Question 13.
Motion of a coin on a carrom board is an example of -
(a) one dimensional motion
(b) two dimensional motion
(c) three dimensional motion
(d) none
Answer:
(b) two dimensional motion
Question 15.
A bird flying in the sky is an example of -
(a) one dimensional motion
(b) two dimensional motion
(c) three dimensional motion
(d) none
Answer:
(c) three dimensional motion
Question 16.
Example for scalar is -
(a) distance
(b) displacement
(c) velocity
(d) angular momentum
Answer:
(a) distance

Question 17.
Which of the following is not a scalar?
(a) Volume
(b) angular momentum
(c) Relative density
(d) time
Answer:
(b) angular momentum
Question 18.
Vector is having -
(a) only magnitude
(b) only direction
(c) bot magnitude and direction
(d) either magnitude or direction
Answer:
(c) both magnitude and direction

Question 19.
"norm" of the vector represents -
(a) only magnitude
(b) only direction
(c) both magnitude and direction
(d) either magnitude or direction
Answer:
(a) only magnitude
Question 20.
If two vectors are having equal magnitude and same direction is known as -
(a) equal vectors
(b) col-linear vectors
(c) parallel vectors
(d) on it vector
Answer:
(a) equal vectors
Question 21.
The angle between two collinear vectors is / are -
(a) $0^{\circ}$
(b) $90^{\circ}$
(c) $180^{\circ}$
(d) $0^{\circ}$ (or) $180^{\circ}$
Answer:
(d) $0^{\circ}$ (or) $180^{\circ}$

(d) $0^{\circ}$ (or) $180^{\circ}$
Question 22.
The angle between parallel vectors is -
(a) $0^{\circ}$
(b) $90^{\circ}$
(c) $180^{\circ}$
(d) $0^{\circ}$ (or) $180^{\circ}$
Answer:
(a) $0^{\circ}$
Question 23.
The angle between anti parallel vectors is -
(a) $0^{\circ}$
(b) $90^{\circ}$
(c) $180^{\circ}$
(d) $0^{\circ}$ (or) $180^{\circ}$
Answer:
(c) $180^{\circ}$

Question 24.
Unit vector is -
(a) having magnitude one but no direction
(b) $A \widehat{A}$
(c) $\frac{\hat{A}}{A}$
(d) $|\mathrm{A}|$
Answer:
(c) $\frac{\hat{A}}{A}$
Question 25 .
A unit vector is used to specify -
(a) only magnitude
(b) only direction
(c) either magnitude (or) direction
(d) absolute value
Answer:
(b) only direction
Question 26.
The angle between any two orthogonal unit vectors is -
(a) $0^{\circ}$
(b) $90^{\circ}$
(c) $180^{\circ}$
(d) $360^{\circ}$
Answer:
(b) $90^{\circ}$

Question 27.
If $\hat{n}$ is a unit vector along the direction of $\overrightarrow{\mathrm{A}}$, the $\hat{n}$ is-
(a) $\overrightarrow{\mathrm{A}} \mathrm{A}$
(b) $\mathrm{n} \times \mathrm{A}$
(c) $\overrightarrow{\mathrm{A}} / \mathrm{A}$
(d $\overrightarrow{\mathrm{A}}|\mathrm{A}|$
Answer:
(c) $\overrightarrow{\mathrm{A}} / \mathrm{A}$
Question 28.
The magnitude of a vector can not be-
(a) positive
(b) negative
(e) zero
(c) 90

Answer:
(b) negative
Question 29.

If $R=P+Q$, then which of the following is true?
(a) $P>Q$
(b) Q $>$ P
(c) $P=Q$
(d) R $>$ P,$Q$
Answer:
(d) R $>$ P, Q
Question 30 .
A force of $3 \mathrm{~N}$ and $4 \mathrm{~N}$ are acting perpendicular to an object, the resultant force is-
(a) $9 \mathrm{~N}$
(b) $16 \mathrm{~N}$
(c) $5 \mathrm{~N}$
(d) $7 \mathrm{~N}$
Answer:
(c) $5 \mathrm{~N}$
Question 31.
Torque is a-
(a) scalar
(b) vector
(c) either scalar (or) vector
(d) none
Answer:
(6) vector

Question 32 .
The resultant of $\overrightarrow{\mathrm{A}}+\overrightarrow{\mathrm{B}}$ acts along $\mathrm{x}-$ axis. If $\mathrm{A}=2 \hat{i}-3 \hat{j}+2 \hat{k}$ then $\mathrm{B}$ is-
(a) $-2 \hat{i}+\hat{j}+\hat{k}$
(b) $3 \hat{j}-2 \hat{k}$
(c) $-2 \hat{i}-3 \hat{j}$
(d) $-2 \hat{i}-2 \hat{k}$
Answer:
(b) $3 \hat{j}-2 \hat{k}$

Question 33.
The angle between $(\vec{A}+\vec{B})$ and $\overrightarrow{(A}-\vec{B})$ can be -
(a) only $0^{\circ}$
(b) only $90^{\circ}$
(c) between $0^{\circ}$ and $90^{\circ}$
(d) between $0^{\circ}$ and $180^{\circ}$
Answer:
(d) between $0^{\circ}$ and $180^{\circ}$
Question 34 .
If a vector $\overrightarrow{\mathrm{A}}=3 \hat{i}+2 \hat{j}$ then what is $4 \mathrm{~A}$ -
(a) $12 \hat{i}+8 \hat{j}$
(b) $0.75 \hat{i}+0.5 \hat{j}$
(c) $3 \hat{i}+2 \hat{j}$
(d) $7 \hat{i}+6 \hat{j}$
Answer:
(a) $12 \hat{i}+8 \hat{j}$
Question 35 .
If $\mathrm{P}=\mathrm{mV}$ then the direction of $\mathrm{P}$ along-
(a) $\mathrm{m}$
(b) $\mathrm{v}$
(c) both (a) and (b)
(d) neither $\mathrm{m}$ nor $\mathrm{v}$
Answer:
(b) $\mathrm{v}$

Question 36.
The scalar product $\overrightarrow{\mathrm{A}} \cdot \overrightarrow{\mathrm{B}}$ is equal to-
(a) $\overrightarrow{\mathrm{A}}+\overrightarrow{\mathrm{B}}$
(b) $\mathrm{AB} \sin \theta$
(c) $\mathrm{AB} \cos \theta$
(d) $\vec{A}+\vec{B}$
Answer:
(c) $\mathrm{AB} \cos \theta$
Question 37.
The scalar product $\vec{A} \cdot \vec{B}$ is equal to-
(a) $\overrightarrow{\mathrm{A}}+\overrightarrow{\mathrm{B}}$
(b) $\overrightarrow{\mathrm{A}} \cdot \overrightarrow{\mathrm{B}}$

(c) $\mathrm{AB} \sin \theta$
(d) $\vec{A} \times \vec{B}$
Answer:
(b) $\overrightarrow{\mathrm{A}} \cdot \overrightarrow{\mathrm{B}}$
Question 38.
The scalar product of two vectors will be maximum when $\theta$ is equal to -
(a) $0^{\circ}$
(b) $90^{\circ}$
(c) $180^{\circ}$
(d) $270^{\circ}$
Answer:
(a) $0^{\circ}$
Question 39.
The scalar product of two vectors will be minimum. When $\theta$ is equal to -
(a) $0^{\circ}$
(b) $45^{\circ}$
(c) $180^{\circ}$
(d) $60^{\circ}$
Answer:
(c) $180^{\circ}$
Question 40.
The vectors A and B to be mutually orthogonal when -
(a) $\vec{A}+\vec{B}=0$
(b) $\overrightarrow{\mathrm{A}}-\overrightarrow{\mathrm{B}}=0$

(c) $\overrightarrow{\mathrm{A}} \cdot \overrightarrow{\mathrm{B}}=0$
(d) $\overrightarrow{\mathrm{A}} \times \overrightarrow{\mathrm{B}}=0$
Answer:
(c) $\overrightarrow{\mathrm{A}} \cdot \overrightarrow{\mathrm{B}}=0$
Question 41.
The magnitude of the vector is -
(a) $\mathrm{A}^2$
(b) $\sqrt{A^2}$
(c) $\sqrt{\mathrm{A}}$
(d) $\sqrt[3]{\mathrm{A}}$
Answer:
(b) $\sqrt{\mathrm{A}^2}$

Question 42.
$\hat{i} \cdot \hat{j}$ is -
(a) 0
(b) I
(c) $\infty$
(d) none
Answer:
(a) 0
Question 43.
If $\overrightarrow{\mathrm{A}}$ and $\overrightarrow{\mathrm{B}}$ are two vectors, which are acting along $x$, y respectively, then $\vec{A}$ and $\vec{B}$ lies along -
(a) $\mathrm{x}$
(b) $\mathrm{y}$
(c) $z$
(d) none
Answer:
(c) $z$
Question 44.
The direction of $\overrightarrow{\mathrm{A}} \times \overrightarrow{\mathrm{B}}$ is given by-
(a) right hand screw rule
(b) right hand thumb rule
(c) both (a) and (b)
(d) neither (a) and (b)
Answer:
(c) both (a) and (b)

Question 45.
$\overrightarrow{\mathrm{A}} \times \overrightarrow{\mathrm{B}}$ is -
(a) $\mathrm{AB} \cos \theta$
(b) $\mathrm{AB} \sin \theta$
(c) $A B \tan \theta$
(d) $A B \sec \theta$
Answer:
(b) $\mathrm{AB} \sin \theta$
Question 46.
$\overrightarrow{\mathrm{A}} \times \overrightarrow{\mathrm{B}}$ isequal to -
(a) $\overrightarrow{\mathrm{B}} \times \overrightarrow{\mathrm{A}}$
(b) $\overrightarrow{\mathrm{A}}+\overrightarrow{\mathrm{B}}$

(c) $-\vec{B} \times \vec{A}$
(d) $\vec{A}-\vec{B}$
Answer:
(c) $-\overrightarrow{\mathrm{B}} \times \overrightarrow{\mathrm{A}}$
Question 47.
The vector product of any two vectors gives a -
(a) vector
(b) scalar
(e) tensor
(d) col-linear
Answer:
(a) vector
Question 48.
$|\overrightarrow{\mathrm{A}} \times \overrightarrow{\mathrm{B}}|$ is equal to -
(a) $-|\vec{A} \times \vec{B}|$
(b) $|\overrightarrow{\mathrm{B}} \times \overrightarrow{\mathrm{A}}|$
(c) $-|\overrightarrow{\mathrm{B}} \times \overrightarrow{\mathrm{A}}|$
(d) $\frac{\overline{\mathrm{A}} \times \overline{\mathrm{B}}}{|\overline{\mathrm{A}} \times \overline{\mathrm{B}}|}$
Answer:
(b) $|\overrightarrow{\mathrm{B}} \times \overrightarrow{\mathrm{A}}|$

Question 49.
The vector product of two vectors will have maximum magnitude when $\theta$ is equal to -
(a) $0^{\circ}$
(b) $90^{\circ}$
(c) $180^{\circ}$
(d) $360^{\circ}$
Answer:
(b) $90^{\circ}$
Question 50 .
The vector product of two non-zero vectors will be minimum when $\mathrm{O}$ is equal to -
(a) $0^{\circ}$
(b) $180^{\circ}$
(e) both (a) and (b)
(d) neither (a) nor (b)
Answer:
(e) both (a) and (b)

Question 49.
The vector product of two vectors will have maximum magnitude when $\theta$ is equal to -
(a) $0^{\circ}$
(b) $90^{\circ}$
(c) $180^{\circ}$
(d) $360^{\circ}$
Answer:
(b) $90^{\circ}$
Question 50 .
The vector product of two non-zero vectors will be minimum when $\mathrm{O}$ is equal to -
(a) $0^{\circ}$
(b) $180^{\circ}$
(e) both (a) and (b)
(d) neither (a) nor (b)
Answer:
(e) both (a) and (b)

Question 54 .
$\hat{j} \mathrm{x} \hat{i}$ is -
(a) $-\hat{i}$
(b) $-\hat{j}$
(c) $-\hat{k}$
(d) $\overrightarrow{\mathrm{z}}$
Answer:
(c) $-\hat{k}$
Question 55.
If two vectors $\vec{A}$ and $\vec{B}$ form adjacent sides of parallelogram, then the magnitude of $|\vec{A} \times \vec{B}|$ will give of parallelogram -
(a) length
(b) area
(c) volume
(d) diagonal
Answer:

(b) area
Question 56.
If $\overrightarrow{\mathrm{P}}-\overrightarrow{\mathrm{Q}}$ then which of the following is incorrect. -
(a) $\hat{P}=\hat{Q}$
(b) $|\hat{P}|=|\hat{Q}|$
(c) $\mathrm{P} \hat{Q}=\mathrm{Q} \hat{A}$
(d) $\hat{P} \hat{Q}=\mathrm{PQ}$
Answer:
(d) $\hat{P} \hat{Q}=\mathrm{PQ}$
Question 57.
The momentum of a particle is $\overrightarrow{\mathrm{P}}=\cos \theta \hat{i}+\sin \theta \hat{j}$. The angle between momentum and the force acting on a body is -
(a) $0^{\circ}$
(b) $45^{\circ}$
(c) $90^{\circ}$
(d) $180^{\circ}$
Answer:
(c) $90^{\circ}$
Question 58.
$\mathrm{A}$ and $\mathrm{B}$ are two vectors, if $\mathrm{A}$ and $\mathrm{B}$ are perpendicular to each other then -
(a) $\overrightarrow{\mathrm{A}} \times \overrightarrow{\mathrm{B}}=0$
(b) $\overrightarrow{\mathrm{A}} \times \overrightarrow{\mathrm{B}}=1$
(c) $\overrightarrow{\mathrm{A}} \overrightarrow{\mathrm{B}}=0$

(d) $\overrightarrow{\mathrm{A}} \overrightarrow{\mathrm{B}}=\overrightarrow{\mathrm{A} \mathrm{B}}$
Answer:
(c) $\overrightarrow{\mathrm{A}} \overrightarrow{\mathrm{B}}=0$
Question 59.
The angle between two vectors $-3 \hat{i}+6 \hat{k}$ and $2 \hat{i}+3 \hat{j}+\hat{k}$ is -
(a) $0^{\circ}$
(b) $45^{\circ}$
(c) $60^{\circ}$
(d) $90^{\circ}$
Answer:
(d) $90^{\circ}$
Question 60.
The radius vector is $2 \hat{i}+\hat{j}+\hat{k}$ while linear momentum is $2 \hat{i}+3 \hat{j}+\hat{k}$ Then the angular momentum is
(a) $-2 \hat{i}+4 \hat{k}$
(b) $4 \hat{i}-8 \hat{k}$

(c) $2 \hat{k}-4 \hat{j}+2 \hat{k}$
(d) $4 \hat{i}-8 \hat{j}$
Answer:
(a) $-2 \hat{i}+4 \hat{k}$
Question 61.
Which of the following cannot be a resultant of two vectors of magnitude 3 and 6 ?
(a) 3
(b) 6
(c) 10
(d) 7
Answer:
(c) 10
Question 62.
Twelve forces each of magnitude $10 \mathrm{~N}$ acting on a body at an angle of $30^{\circ}$ with other forces then their resultant is-
(a) $10 \mathrm{~N}$
(b) $120 \mathrm{~N}$
(c) $\frac{10}{\sqrt{3}}$
(d) zero
Question 63.
Two forces are in the ratio of $3: 4$. The maximum and minimum of their resultants are in the ratio is -
(a) $4: 3$
(b) $3: 4$
(c) $7: 1$
(d) $1: 7$

Answer:
(c) $7: 1$
Question 64.
If $|\overrightarrow{\mathrm{P}}+\overrightarrow{\mathrm{Q}}|=|\overrightarrow{\mathrm{P}}|+|\overrightarrow{\mathrm{Q}}|$. The angle between the vectors $\overrightarrow{\mathrm{P}}$ and $\overrightarrow{\mathrm{Q}}$ is -
(a) $0^{\circ}$
(b) $180^{\circ}$
(c) $60^{\circ}$
(d) $90^{\circ}$
Answer:
(a) $0^{\circ}$
$
|\overrightarrow{\mathrm{P}}+\overrightarrow{\mathrm{Q}}|=|\overrightarrow{\mathrm{P}}|+|\overrightarrow{\mathrm{Q}}|
$
Square on both sides and the resultant becomes
$
\begin{aligned}
& \mathrm{P}^2+\mathrm{Q}^2+2 \mathrm{PQ} \cos \theta=\mathrm{P}^2+\mathrm{Q}^2+2 \mathrm{PQ} \cos \theta=1 \\
& \theta=0
\end{aligned}
$

Question 65.
If $|\overrightarrow{\mathrm{P}}+\overrightarrow{\mathrm{Q}}=| \overrightarrow{\mathrm{P}}|-| \overrightarrow{\mathrm{P}} \mid$, then the angle between the vectors $\overrightarrow{\mathrm{P}}$ and $\overrightarrow{\mathrm{Q}}$
(a) $0^{\circ}$
(b) $90^{\circ}$
(c) $180^{\circ}$
(d) $360^{\circ}$
Answer:
(c) $|\overrightarrow{\mathrm{P}}+\overrightarrow{\mathrm{Q}}|=|\overrightarrow{\mathrm{P}}||\overrightarrow{\mathrm{P}}|$,
Square on both side, and the resultant becomes
$
\begin{aligned}
& \mathrm{P}^2+\mathrm{Q}^2+2 \mathrm{PQ} \cos \theta=\mathrm{P}^2+\mathrm{Q}^2-2 \mathrm{PQ} . \\
& \cos \theta=-1 \\
& \theta=180^{\circ}
\end{aligned}
$
Question 66.
If $|\overrightarrow{\mathrm{P}} \times \overrightarrow{\mathrm{Q}}|=|\overrightarrow{\mathrm{P}} \cdot \overrightarrow{\mathrm{Q}}|$ then angle between $\overrightarrow{\mathrm{P}}$ and $\overrightarrow{\mathrm{Q}}$ then angle between $\mathrm{P}$ and $\mathrm{Q}$ will be -
(a) $0^{\circ}$
(b) $30^{\circ}$
(c) $45^{\circ}$
(d) $60^{\circ}$
Answer:
(c) $|\overrightarrow{\mathrm{P}} \times \overrightarrow{\mathrm{Q}}|=|\overrightarrow{\mathrm{P}} \cdot \overrightarrow{\mathrm{Q}}|$ Expand the terms
PQ $\sin \theta=P \mathrm{P} \cos \theta$
$\tan \theta=1$
$
\theta=45^{\circ}
$

Question 65.
If $|\overrightarrow{\mathrm{P}}+\overrightarrow{\mathrm{Q}}=| \overrightarrow{\mathrm{P}}|-| \overrightarrow{\mathrm{P}} \mid$, then the angle between the vectors $\overrightarrow{\mathrm{P}}$ and $\overrightarrow{\mathrm{Q}}$
(a) $0^{\circ}$
(b) $90^{\circ}$
(c) $180^{\circ}$
(d) $360^{\circ}$
Answer:
(c) $|\overrightarrow{\mathrm{P}}+\overrightarrow{\mathrm{Q}}|=|\overrightarrow{\mathrm{P}}||\overrightarrow{\mathrm{P}}|$,
Square on both side, and the resultant becomes
$
\begin{aligned}
& \mathrm{P}^2+\mathrm{Q}^2+2 \mathrm{PQ} \cos \theta=\mathrm{P}^2+\mathrm{Q}^2-2 \mathrm{PQ} . \\
& \cos \theta=-1 \\
& \theta=180^{\circ}
\end{aligned}
$
Question 66.
If $|\overrightarrow{\mathrm{P}} \times \overrightarrow{\mathrm{Q}}|=|\overrightarrow{\mathrm{P}} \cdot \overrightarrow{\mathrm{Q}}|$ then angle between $\overrightarrow{\mathrm{P}}$ and $\overrightarrow{\mathrm{Q}}$ then angle between $\mathrm{P}$ and $\mathrm{Q}$ will be -
(a) $0^{\circ}$
(b) $30^{\circ}$
(c) $45^{\circ}$
(d) $60^{\circ}$
Answer:
(c) $|\overrightarrow{\mathrm{P}} \times \overrightarrow{\mathrm{Q}}|=|\overrightarrow{\mathrm{P}} \cdot \overrightarrow{\mathrm{Q}}|$ Expand the terms
PQ $\sin \theta=P \mathrm{P} \cos \theta$
$\tan \theta=1$
$
\theta=45^{\circ}
$

Question 68.
If $A$ and $B$ are the sides of triangle, then area of triangle -
(a) $\frac{1}{2}|\overrightarrow{\mathrm{A}} \cdot \overrightarrow{\mathrm{B}}|$
(b) $\frac{1}{2}|\overrightarrow{\mathrm{A}} \times \overrightarrow{\mathrm{B}}|$
(c) $\mathrm{AB} \sin \theta$
(d) $A B \cos \theta$
Answer:
(b) $\frac{1}{2}|\overrightarrow{\mathrm{A}} \times \overrightarrow{\mathrm{B}}|$
Question 69.
A particle moves in a circular path of radius $2 \mathrm{~cm}$. If a particle completes 3 rounds, then the distance and displacement of the particle are -
(a) 0 and 37.7
(b) 37.7 and 0
(c) 0 and 0
(d) 37.7 and 37.7
Answer:
(b) Radius $=2 \mathrm{~cm}$
Circumference of the circle $=2 \mathrm{nr}=4 \mathrm{n} \mathrm{cm}$
Distance covered in 3 rounds $=127 \mathrm{r} \mathrm{cm}=37.7 \mathrm{~cm}$
Initial and final positions are same
$\therefore$ Displacement $=0$

Question 70.
If $r \mathrm{x}$ and $\mathrm{r} 2$ are position vectors, then the displacement vector is -
(a) $\vec{r}_1 \times \vec{r}_2$
(b) $\vec{r}_1 \cdot \vec{r}_2$
(c) $\vec{r}_1+\overrightarrow{\vec{r}}_2$
(d) $\vec{r}_2+\vec{r}_1$
Answer:
(d) $\vec{r}_2+\vec{r}_1$
Question 71.
The ratio of the displacement vector to the corresponding time interval is -
(a) average speed
(b) average velocity
(c) instantaneous speed
(d) instantaneous velocity
Answer:
(b) average velocity

Question 72.
The ratio of total path length travelled by the particle in a time interval -
(a) average speed
(b) average velocity
(c) instantaneous speed
(d) instantaneous velocity
Answer:
(a) average speed
Question 73.
The product of mass and velocity of a particle is -
(a) acceleration
(b) force
(c) torque
(d) momentum
Answer:
(d) momentum
Question 74.
The area under the force, displacement curve is -
(a) potential energy
(b) work done.
(c) impulse
(d) acceleration
Answer:
(b) work done

Question 75.
The area under the force, time graph is -
(a) momentum
(b) force
(c) work done
(d) impulse
Answer:
(d) impulse
Question 76.
The unit of momentum is -
(a) $\mathrm{kg} \mathrm{ms}^{-1}$
(b) $\mathrm{kg} \mathrm{ms}^{-2}$
(c) $\mathrm{kg} \mathrm{m}^2 \mathrm{~s}^{-1}$
(d) $\mathrm{kg}^{-1} \mathrm{~m}^2 \mathrm{~s}^{-1}$
Answer:
(b) $\mathrm{kg} \mathrm{ms}^{-2}$

Question 77.
The slope of the position - time graph will give -
(a) displacement
(b) velocity
(c) acceleration
(d) force
Answer:
(d) force
Question 78.
The area under velocity-time graph gives-
(a) positive
(b) negative
(c) either positive (or) negative
(d) zero
Answer:
(c) either positive (or) negative
Question 79.
The magnitude of distance is always-
(a) positive
(b) negative
(c) either positive (or) negative
(d) zero
Answer:
(a) positive

Question 80 .
If two objects $A$ and $B$ are moving along a straight line in the same direction with the velocities $v_A$ and $v_B$ respectively, then the relative velocity is-
(a) $v_A+v_B$
(b) $v_A-v_B$
(c) $\mathrm{v}_{\mathrm{A}} \mathrm{v}_{\mathrm{B}}$
(d) $\mathrm{v}_{\mathrm{A}} / \mathrm{v}_{\mathrm{B}}$
Answer:
(b) $\mathrm{V}_{\mathrm{A}}-\mathrm{V}_{\mathrm{B}}$
Question 81.
If two objects $A$ and $B$ are moving along a straight line in the opposite direction with the velocities $V_A$ and $V_B$ respectively, then relative velocity is-
(a) $V_A+V_B$
(b) $\mathrm{V}_{\mathrm{A}}-\mathrm{V}_{\mathrm{B}}$
(c) $V_A \cdot V_B$
(d) $V_A / V_B$

Answer:
(a) $\mathrm{V}_{\mathrm{A}}+\mathrm{V}_{\mathrm{B}}$
Question 82.
If two objects moving with a velocities of $\mathrm{V}_{\mathrm{A}}$ and $\mathrm{V}_{\mathrm{B}}$ at an angle of 0 between them, the relative velocity is -
(a) $\mathrm{V}_{\mathrm{AB}}=\sqrt{\mathrm{V}_{\mathrm{A}}^2+\mathrm{V}_{\mathrm{B}}^2-2 \mathrm{~V}_{\mathrm{A}} \mathrm{V}_{\mathrm{B}} \cos \theta}$
(b) $\mathrm{V}_{\mathrm{AB}}=\sqrt{\mathrm{V}_{\mathrm{A}}^2+\mathrm{V}_{\mathrm{B}}^2+2 \mathrm{~V}_{\mathrm{A}} \mathrm{V}_{\mathrm{B}} \cos \theta}$
(c) $\mathrm{V}_{\mathrm{AB}}=\mathrm{V}_{\mathrm{A}}^2+\mathrm{V}_{\mathrm{B}}^2$
(d) $\mathrm{V}_{\mathrm{AB}}=\mathrm{V}_{\mathrm{A}} \mathrm{V}_{\mathrm{B}} \cos \theta$
Answer:
(a) $\mathrm{V}_{\mathrm{AB}}=\sqrt{\mathrm{V}_{\mathrm{A}}^2+\mathrm{V}_{\mathrm{B}}^2-2 \mathrm{~V}_{\mathrm{A}} \mathrm{V}_{\mathrm{B}} \cos \theta}$
Question 83.
A person moving horizontally with velocity $\overrightarrow{\mathrm{V}_m}$ The relative velocity of rain with respect to the person is -
(a) $V_R+V_m$
(b) $\sqrt{\mathrm{V}_{\mathrm{R}}+\mathrm{V}_m}$
(c) $\mathrm{V}_{\mathrm{R}}-\mathrm{V}_m$
(d) $\sqrt{\mathrm{v}_{\mathrm{R}}^2+\mathrm{V}_m^2}$
Answer:
(d) $\sqrt{\mathrm{v}_{\mathrm{R}}^2+\mathrm{V}_m^2}$
Question 84.
A person moving horizontally with velocity $\overrightarrow{\mathrm{V}_m}$. Rain falls vertically with velocity $\overrightarrow{\mathrm{V}_R}$ To save himself from the rain, he should hold an umbrella with vertical at an angle of -
(a) $\tan ^{-1}\left(\frac{V_R}{V_m}\right)$
(b) $\tan ^{-1}\left(\frac{V_m}{V_R}\right)$
(c) $\tan \theta=V_m+\mathrm{V}_{\mathrm{R}}$
(d) $\tan ^{-1}\left(V_{\mathrm{R}}+\mathrm{V}_m / \mathrm{V}_{\mathrm{R}}-\mathrm{V}_m\right)$
Answer:
(b) $\tan ^{-1}\left(\frac{V_m}{V_R}\right)$

Question 85 .
A car starting from rest, accelerates at a constant rate $\mathrm{x}$ for sometime after which it decelerates at a constant rate $\mathrm{v}$ to come to rest. If the total time elapsed is $t$, the maximum velocity attained by the car is given by -
(a) $\frac{x y}{x+y} t$
(b) $\frac{x y}{x-y} t$
(c) $\frac{x^2 y^2}{x^2+y^2} t$
(d) $\frac{x^2 y^2}{x^2-v^2} t$
Answer:
(a) $\frac{x y}{x+y} t$
Question 86.
A car covers half of its journey with a speed of $10 \mathrm{~ms}^{-1}$ and the other half by $20 \mathrm{~ms}^{-1}$. The average speed of car during the total journey is -
(a) $70 \mathrm{~ms}^{-1}$
(b) $15 \mathrm{~ms}^{-1}$
(c) $13.33 \mathrm{~ms}^{-1}$
(d) $7.5 \mathrm{~ms}^{-1}$

Answer:
(c) Let $\mathrm{x}$ is the total distance
Time to cover 1 st half $=\frac{x / 2}{10}$
Time to cover 2 nd half $=\frac{x / 2}{20}$
Average speed $=$
$
\frac{x}{\frac{x}{20}+\frac{x}{40}}=\frac{1}{\left(\frac{3}{40}\right)}=13.33 \mathrm{~ms}^{-1}
$
Question 87.
A swimmer can swim in still water at of $10 \mathrm{~ms}^{-1}$ While crossing a river his average speed is 6 $\mathrm{ms}^{-1}$. If he crosses the river in the shortest possible time, what is the speed of flow of water?
(a) $16 \mathrm{~ms}^{-1}$

(b) $4 \mathrm{~ms}^{-1}$
(c) $60 \mathrm{~ms}^{-1}$
(d) $8 \mathrm{~ms}^{-1}$
Answer:
(d) The resultant velocity of swimmer must be perpendicular to speed of water to cross the river in a shortest time
$
\begin{aligned}
& \therefore v_s^2=v^2+v_w^2 \\
& v_w^2=v_s^2-v^2=100-36=64 \\
& \therefore \mathrm{V}=8 \mathrm{~m} / \mathrm{s}^{-1}
\end{aligned}
$
Question 88.
A $100 \mathrm{~m}$ long train is traveling from North to South at a speed of $30 \mathrm{~ms}^{-1}$. A bird is flying from South to North at a speed of $10^{-1}$. How long will the bird take to, cross the train?
(a) $3 \mathrm{~s}$
(b) $2.5 \mathrm{~s}$
(c) $10 \mathrm{~s}$
(d) $5 \mathrm{~s}$
Answer:
(b) Length of train $=100 \mathrm{~m}$
Relative velocity $=30+10=40 \mathrm{~ms}^{-1}$
Time taken to cross the train $(\mathrm{t})=\frac{\text { distance }}{\text { R.velocity }}=\frac{100}{40}=2.5 \mathrm{~s}$
Question 89.
The first derivative of position vector with respect to time is -
(a) velocity
(b) acceleration
(c) force
(d) displacement

Answer:
(a) velocity
Question 90 .
The second derivative of position vector with respect to time is -
(a) velocity
(b) acceleration
(c) force
(d) displacement
Answer:
(b) acceleration
Question 91.
The slope of displacement-time graph gives -
(a) velocity
(b) acceleration
(c) force
(d) displacement
Answer:
(a) velocity

Question 92 .
The slope of velocity-time graph gives -
(a) velocity
(b) acceleration
(c) force
(d) displacement
Answer:
(b) acceleration
Question 93.
The position vector of a particle is $\vec{r}=4 \mathrm{t}^2 \hat{i}+2 \mathrm{t} \hat{j}+3 \mathrm{t} \hat{k}$ The acceleration of a particle is having only -
(a) X-component
(b) Y-component
(c) $\mathrm{Z}$ - component
(d) $\mathrm{X}-\mathrm{Y}$ component
Answer:
(a) $\mathrm{X}$ - component
$4 \mathrm{t}^2 \hat{i}+2 \mathrm{t} \hat{j}+3 \mathrm{t} \hat{k}$
$\vec{v}=\frac{d \vec{r}}{d t}=8 \mathrm{t} \hat{i}+2 \hat{j}$
$\mathrm{a}=\frac{d^2 r}{d t^2}=8 \hat{i}$ a is having only $\mathrm{X}$-component.
Question 94.
The position vector of a particle is $\vec{r}=4 \mathrm{t}^2 \hat{i}+2 \mathrm{t} \hat{j}+3 \mathrm{t} \hat{k}$. The speed of the particle at $\mathrm{t}=5 \mathrm{~s}$ is -
(a) $42 \mathrm{~ms}^{-1}$
(b) $3 \mathrm{~s}$
(c) $3 \mathrm{~ms}^{-1}$

(d) $40 \mathrm{~ms}^{-1}$
Answer:
(a) $42 \mathrm{~ms}^{-1}$
$\vec{r}=4 \mathrm{t}^2 \hat{i}+2 \mathrm{t} \hat{j}+3 \mathrm{t} \hat{k}$
Speed $\mathrm{v}=-\frac{d \vec{r}}{d t}=8 \mathrm{t} \hat{i}+2 \hat{j}$
at $\mathrm{t}=5 \mathrm{~s} \mathrm{v}=40+2=42$
Question 95.
An object is moving in a straight line with uniform acceleration a, the velocity-time relation is
(a) $u=v+a t$
(b) $\mathrm{v}=\mathrm{u}+$ at
(c) $v^2=u^2+a^2 t^2$
(d) $v^2-u^2=a t$
Answer:
(b) $\mathrm{v}=\mathrm{u}+$ at

Question 96.
An object is moving in a straight line with uniform acceleration, the displacement-time relation is -
(a) $\mathrm{S}=u t^2+\frac{1}{2} a t^2$
(b) $\mathrm{S}=u t-\frac{1}{2} a t^2$
(c) $\mathrm{S}=u t+\frac{1}{2} a t^2$
(d) $\mathrm{S}=u t-a t^2$
Answer:
(c) $\mathrm{S}=u t+\frac{1}{2} a t^2$
Question 97.
An object is moving in a straight line with uniform acceleration, the velocity-displacement reflation is -
(a) $\mathrm{V}=\mathrm{u}+2 \mathrm{as}$
(b) $\mathrm{S}=\mathrm{ut}+$-at
(c) $\mathrm{V}^2=\mathrm{u}^2-2 \mathrm{as}$
(d) $\mathrm{V}^2=u^2+2 \mathrm{as}$
Answer:
(d) $\mathrm{V}^2=u^2+2 \mathrm{as}$
Question 98 .
For free-falling body, its initial velocity is -
(a) 0
(b) 1
(c) $\infty$
(d) none
Answer:
(a) 0

Question 99.
An object falls from a height $\mathrm{h}(\mathrm{h}<<\mathrm{R})$, the speed of the object when it reaches the ground is -
(a) $\frac{1}{2} g t^2$
(b) $\sqrt{g t}$
(c) gh
(d) $\sqrt{2 g h}$
Answer:
(d) $\sqrt{2 g h}$

Question 100 .
An object falls from a height $\mathrm{h}(\mathrm{h}<<\mathrm{R})$ the time taken by an object to reaches the ground is -
(a) $\frac{1}{2} g t^2$
(b) $\sqrt{2 g h}$
(c) $\sqrt{\frac{2 h}{g}}$
(d) $\sqrt{\frac{2 g}{h}}$
Answer:
(d) $\sqrt{\frac{2 g}{h}}$
Question 101.
In the absence of air resistance, horizontal velocity of the projectile is -
(a) always negative
(b) equal to ' $\mathrm{g}$ '
(c) directly proportional to $g$
(d) a constant
Answer:
(d) a constant
Question 102.
In the horizontal projection, the range of the projectile is -
(a) $\sqrt{\frac{2 h}{g}}$
(b) $u \sqrt{\frac{h}{g}}$
(c) $u \sqrt{\frac{2 h}{g}}$
(d) $u \sqrt{\frac{g}{2 h}}$

Answer:
(c) $u \sqrt{\frac{2 h}{g}}$
Question 103.
In oblique projection, maximum height attained by the projectile is -
(a) $\frac{t}{u \cos \theta}$
(b) $\frac{u \cos \theta}{2 g}$
(c) $\frac{2 g}{u \cos \theta}$
(d) $\frac{u^2 \sin ^2 \theta}{2 g}$
Answer:
(d) $\frac{u^2 \sin ^2 \theta}{2 g}$

Question 104.
In oblique projection time of flight of a projectile is -
(a) $\frac{u^2 \sin ^2 \theta}{2 g}$
(b) $\frac{2 u \cos \theta}{g}$
(c) $\frac{u^2 \sin 2 \theta}{g}$
(d) $\frac{u^2}{g}$
Answer:
(b) $\frac{2 u \cos \theta}{g}$
Question 105.
In oblique projection horizontal range of the projectile is -
(a) $\frac{u^2 \sin ^2 \theta}{2 g}$
(b) $\frac{2 u \cos \theta}{g}$
(c) $\frac{u^2 \sin 2 \theta}{g}$
(d) $\frac{u^2}{g}$
Answer:
(a) $\frac{u^2 \sin ^2 \theta}{2 g}$
Question 106.
In oblique projection, maximum horizontal range of the projectile is -
(a) $\frac{u^2 \sin ^2 \theta}{2 g}$
(b) $\frac{2 u \cos \theta}{g}$
(c) $\frac{u^2 \sin 2 \theta}{g}$

(d) $\frac{u^2}{g}$
Answer:
(d) $\frac{u^2}{g}$
Question 107.
One radian is equal to -
(a) $\frac{\pi}{180}$ degree
(b) $60^{\circ}$
(c) $57.295^{\circ}$
(d) $53.925^{\circ}$
Answer:
(c) $57.295^{\circ}$
Question 108.
In relation between linear and angular velocity is -

(a) $\omega=\mathrm{vr}$
(b) $\omega=\frac{v}{r}$
(c) $\omega=\frac{r}{v}$
(d) $\omega=\frac{r}{\omega}$
Answer:
(b) $\omega=\frac{v}{r}$
Question 109.
Centripetal acceleration is given by -
(a) $\frac{v^2}{r}$
(b) $-\frac{v^2}{r}$
(c) $\frac{r}{v^2}$
(d) $-\frac{r}{v^2}$
Answer:
(b) $-\frac{v^2}{r}$
Question 110.
In uniform circular motion -
(a) Speed changes but velocity constant
(b) Velocity changes but speed constant
(c) both speed and velocity are constant
(d) both speed and velocity are variable
Answer:
(b) Velocity changes but speed constant
Question 111.
In non - uniform circular motion, the resultant acceleration is given by -

(a) $a_R=\sqrt{a_t^2-\left(\frac{\mathrm{V}^2}{r}\right)^2}$
(b) $a_R=\sqrt{a_t^2+\left(\frac{\mathrm{V}^2}{r}\right)^2}$
(c) $a_R=\sqrt{a_t^2-\left(\frac{r}{\mathrm{v}^2}\right)^2}$
(d) $a_R=\sqrt{a_t^2+\left(\frac{r}{\mathrm{~V}^2}\right)^2}$

Answer:
(b) $a_R=\sqrt{a_t^2+\left(\frac{\mathrm{V}^2}{r}\right)^2}$
Question 112.
In non - uniform circular motion, the resultant acceleration makes an angle with the radius vector is -
(a) $\tan ^{-1}\left(\frac{r a_t}{v^2}\right)$
(b) $\tan ^{-1}\left(\frac{a_t}{\left(\frac{r}{v^2}\right)}\right)$
(c) $\tan ^{-1}\left(\frac{r v^2}{a t}\right)$
(d) $\tan ^{-1}\left(\frac{r+a t^2}{v^2}\right)$
Answer:
(a) $\tan ^{-1}\left(\frac{r a_t}{v^2}\right)$

Question 113.
A compartment of an uniformly moving train is suddenly detached from the train and stops after covering some distance. The distance covered by the compartment and distance covered by the train in the given time -
(a) both will be equal
(b) second will be half of first
(c) first will be half of second
(d) none
Answer:
(c) first will be half of second
Question 114 .
An object is dropped from rest. Its $\mathrm{v}-\mathrm{t}$ graph is -

Answer:

Question 115.
When a ball hits the ground as free fall and renounces but less than its original height? Which is represented by -

Answer:

Question 116.
Which of the following graph represents the equation $\mathrm{y}=\mathrm{mx}-\mathrm{C}$ ?

Answer:

Question 117.
Which of the following graph represents the equation $\mathrm{y}=\mathrm{mx}+\mathrm{C}$ ?

Answer:

Question 118 .
Which of the following graph represents the equation $y=m x$ ?

Answer:

Question 119.
Which of the following graph represents the equation $y=-m x+C$ ?

Answer:

Question 120 .
Which of the following graph represents the equation $\mathrm{y}=\mathrm{kx}^2$ ?

Answer:

Question 121 .
$
\mathrm{X}=-\mathrm{ky}^2 \text { is represented by }-
$

Answer:

(c)

Question 122.
$\mathrm{X}=\mathrm{ky}^2$ is represented by -

Answer:

Question 123 .
$\mathrm{y}=\mathrm{kx}^2$ is represented by -

Answer:

Question 124 .
$\mathrm{X}^{\circ} \propto \frac{1}{Y}$ (or) $\mathrm{XY}=$ constant is represented by -

Answer:

Question 125 .
$\mathrm{y}=\mathrm{e}^{-\mathrm{kx}}$ is represented by -

Answer:

Question 126.
$
\mathrm{Y}=1-\mathrm{e}^{-\mathrm{kx}} \text { is represented by }-
$

Answer:

Question 127.
$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is represented by -

Answer:

Question 128 .
Let $y=f(x)$ is a function. Its maxima (or) minima can be obtained by -
(a) $y=0$
(b) $f(x)=0$
(c) $\frac{d y}{d x}=0$
(d) $\frac{d^2 y}{d x^2}=0$
Answer:
(c) $\frac{d y}{d x}=0$
Question 129.
A particle at rest starts moving in a horizontal straight line with uniform acceleration. The ratio of the distance covered during the fourth and the third second is -
(a) $\frac{4}{3}$
(b) $\frac{26}{9}$
(e) $\frac{7}{5}$
(d) 2

(c) The distance travelled during nth second -
$
\mathrm{S}_{\mathrm{n}}=\mathrm{u}+\frac{1}{2} \mathrm{a}(2 \mathrm{n}-1)
$
Distance travelled during 4 th second $\mathrm{S}_1=\frac{1}{2}(8-1)$
Distance travelled during 3rd second $\mathrm{S}_2=\frac{1}{2} \mathrm{a}(6-1)$
$
\frac{\mathrm{S}_1}{\mathrm{~S}_2}=\frac{7}{5}
$
Question 130.
The distance travelled by a body, falling freely from rest in $\mathrm{t}=1 \mathrm{~s}, \mathrm{t}=2 \mathrm{~s}$ and $t=3 \mathrm{~s}$ are in the ratio of -
(a) $1: 2: 3$
(h) $1: 3: 5$
(c) $1: 4: 9$
(d) $9: 4: 1$
Answer:
(c) The distance travelled by a free falling body $\mathrm{S}=\frac{1}{2} \mathrm{gt}^2$
$\therefore \mathrm{S} \alpha \mathrm{t}^2$
$\therefore \mathrm{S}_1: \mathrm{S}_2: \mathrm{S}_3: 1^2: 2^2: 3^2=1: 4: 9$.
Question 131.
The displacement of the particle along a straight line at time ; is given by $\mathrm{X}=\mathrm{a}+\mathrm{ht}+\mathrm{ct}^2$ where $a, b, c$ are constants. The acceleration of the particle is-
(a) a
(b) $b$
(c) $\mathrm{c}$
(d) $2 \mathrm{c}$
Answer:

(d) $\mathrm{X}=\mathrm{a}+\mathrm{bt}+\mathrm{ct}{ }^2$
$\frac{d X}{d t}=\mathrm{v}=\mathrm{b}+2 \mathrm{ct}$
Acceleration $=\frac{d^2 X}{d t^2}=2 \mathrm{c}$.
Question 132.
Two bullets are fired at an angle of $\theta$ and $(90-\theta)$ to the horizontal with same speed. The ratio of their times of flight is -
(a) $1: 1$
(b) $1: \tan \theta$
(c) $\tan \theta: 1$
(d) $\tan ^2 \theta: 1$
Answer:
(c) Time of flight $\mathrm{t}_{\mathrm{f}}=\frac{2 x \sin \theta}{9}$
$\mathrm{t}_{\mathrm{f}} \alpha \sin \theta$
$
\begin{aligned}
& \therefore \frac{t_{f_1}}{t_{f_2}}=\frac{\sin \theta}{\sin (90-\theta)}=\frac{\sin \theta}{\cos \theta}=\tan \theta \\
& t_{f_1}: t_{f_2}=\tan \theta: 1
\end{aligned}
$
Question 133.
A particle moves along a circular path under the action of a force. The work done by the force is -
(a) positive and non zero
(b) zero
(c) negative and non-zero
(d) none
Answer:
(b) zero

Question 134.
For a particle, revolving in a circle with speed, the acceleration of the particle is -
(a) along the tangent
(b) along the radius
(c) along its circumference
(d) zero
Answer:
(b) along the radius
Question 135 .
A gun fires two bullets with same velocity at $60^{\circ}$ and $30^{\circ}$ with horizontal. The bullets strike at the same horizontal distance. The ratio of maximum height for the two bullets is in the ratio of -
(a) $1: 2$
(b) $3: 1$
(c) $2: 1$
(d) $1: 3$
Answer:
(b) $3: 1$

$
\begin{aligned}
& \text { Max height attained } \mathrm{h}_{\max }=\frac{u^2 \sin ^2 \theta}{2 g} \\
& \therefore \mathrm{h}_{\max } \alpha \sin ^2 \theta \text { i.e } \mathrm{h}_{\max } \alpha \frac{1-\cos 2 \theta}{2} \\
& \frac{h_{\max 1}}{h_{\max } 2}=\frac{3 / 2}{1 / 2}=3
\end{aligned}
$
Question 136.
A ball is thrown vertically upward. it is a speed of lo $\mathrm{m} / \mathrm{s}$. When it has reached one half of its maximum height. I-low high does the ball rise? $\left(\mathrm{g}=10 \mathrm{~ms}^{-2}\right)$
(a) $5 \mathrm{~m}$
(b) $7 \mathrm{~m}$
(c) $10 \mathrm{~m}$
(d) $12 \mathrm{~m}$
Answer:
(c) $10 \mathrm{~m}$
Question 137.
A car moves from $\mathrm{X}$ to $\mathrm{Y}$ with a uniform speed $\mathrm{V}_{\mathrm{n}}$ and returns to $\mathrm{Y}$ with a uniform speed $\mathrm{V}_{\mathrm{d}}$ The average speed for this round trip is -
(a) $\sqrt{v_u v_d}$
(b) $\frac{v_u v_d}{v_d+v_u}$
(c) $\frac{v_u+v_d}{2}$

(d) $\frac{2 v_d v_u}{v_d+v_u}$
Answer:
(d) $\frac{2 v_d v_u}{v_d+v_u}$
Question 138.
Two projectiles of same mass and with same velocity are thrown at an angle of $60^{\circ}$ and $30^{\circ}$ with the horizontal then which of the following will remain same?
(a) time of flight
(b) range of projectile
(c) maximum height reached
(d) all the above
Answer:
(b) range of projectile

Question 139.
A $\mathrm{n}$ object of mass $3 \mathrm{~kg}$ is at rest. Now a force of $\overrightarrow{\mathrm{F}}=6 \mathrm{t}^2 \hat{i}+4 \mathrm{t} \hat{j}$ is applied on the object, then the velocity of object at $\mathrm{t}=3$ second is -
(a) $18 \hat{i}+3 \hat{j}$
(b) $18 \hat{i}+6 \hat{j}$
(c) $3 \hat{i}+18 \hat{j}$
(d) $18 \hat{i}+4 \hat{j}$
Answer:
(b) $\mathrm{F}=6 \mathrm{t}^2 \hat{i}+4 \mathrm{t} \hat{j}$
$
\begin{aligned}
\mathrm{F}=m a \quad \therefore a=\frac{\overline{\mathrm{F}}}{m}=\frac{1}{3}\left(6 t^2 \hat{i}+4 t \hat{j}\right)=2 t^2 \hat{i}+\frac{4}{2} t \hat{j} \\
a=\frac{d \mathrm{~V}}{d t} \quad \therefore v=\int_0^t a d t=\int_0^3 2 t^2 \hat{i} d t+\int_0^3 \frac{4}{3} t \hat{j} d t=18 \hat{i}+6 \hat{j}
\end{aligned}
$
Question 140.
The angle for which maximum height and horizontal range are same for a projectile is -
(a) $32^{\circ}$
(b) $48^{\circ}$
(c) $76^{\circ}$
(d) $84^{\circ}$
Answer:
(c) $76^{\circ}$
$\mathrm{H}_{\max }=$ horizontal range
$\frac{u^2 \sin ^2 \theta}{2 g}=\frac{u^2 \sin 2 \theta}{g}$

$
\frac{\sin ^2 \theta}{2}=2 \sin \theta \cos \theta=\sin \theta=4 \cos \theta \tan \theta=4 \theta=76^{\circ}
$
Question 141.
A bullet is dropped from some height, when another bullet is fired horizontally from the same height. They will hit the ground -

(a) depends upon mass of bullet
(b) depends upon the observer
(c) one after another
(d) simultaneously
Answer:
(d) simultaneously
Question 142.
From this velocity - time graph, which of the following is correct?
(a) Constant acceleration
(b) Variable acceleration
(c) Constant velocity
(d) Variable acceleration
Answer:
(b) Variable acceleration
Question 143.
When a projectile is at its maximum height, the direction of its velocity and acceleration are -
(a) parallel to each other
(b) perpendicular to each other
(c) anti - parallel to each other
(d) depends on its speed
Answer:
(b) perpendicular to each other

Question 144.
At the highest point of oblique projection, which of the following is correct?
(a) velocity of the projectile is zero
(b) acceleration of the projectile is zero
(c) acceleration of the projectile is vertically downwards
(d) velocity of the projectile is vertically downwards
Answer:
(c) acceleration of the projectile Is vertically downwards
Question 145.
The range of the projectile depends -
(a) The angle of projection
(b) Velocity of projection
(c) $g$
(d) all the above
Answer:
(d) all the above

Question 146.
A constant force is acting on a particle and also acting perpendicular to the velocity of the particle. The particle describes the motion in a plane. Then -
(a) angular displacement is zero
(b) its velocity is zero
(c) it velocity is constant
(d) it moves in a circular path
Answer:
(d) it moves in a circular path
Question 147.
If a body moving in a circular path with uniform speed, then -
(a) the acceleration is directed towards its center
(b) velocity and acceleration are perpendicular to each other
(c) speed of the body is constant but its velocity is varying
(d) all the above
Answer:
(d) all the above
Question 148.
A body is projected vertically upward with the velocity $\mathrm{y}=3 \hat{i}+4 \hat{j} \mathrm{~ms}^{-1}$. The maximum height attained by the body is $\left(\mathrm{g} 10 \mathrm{~ms}^{-2}\right)$.
(a) $7 \mathrm{~m}$
(b) $1.25 \mathrm{~m}$
(c) $8 \mathrm{~m}$
(d) $0.08 \mathrm{~m}$
Answer:
(b) $\mathrm{v}=3 \hat{i}+4 \hat{j}$

$
\begin{aligned}
& \mathrm{H}_{\max }=\frac{v^2 \sin ^2 \theta}{2 g}=\frac{v^2}{2 g}[\theta=90] \\
& \mathrm{v}=\sqrt{9+16}=\sqrt{25} \\
& \mathrm{v}^2=25 \\
& \mathrm{H}_{\max }=\frac{25}{20}=\frac{5}{4}=1.25 \mathrm{~m}
\end{aligned}
$
Short Answer Questions - I (1 Mark)
Question 1.
What is frame of reference?
Answer:
In a coordinate system, the position of an object is described relative to it, then such a coordinate system is called as frame of reference.
Question 2 .
What are the types of motion?
Answer:

- Linear motion
- Circular motion
- Rotational motion
- Vibratory motion.
Question 3.
What is linear motion? Give example.
Answer:
An object is said to be in linear motion if it moves in a straight line.
Example - an athlete running on a straight track.
Question 4.
What is circular motion? Give example.
Answer:
Circular motion is defined as a motion described by an object traversing a circular path. Example - The whirling motion of a stone attached to a string.
Question 5.
What is rotational motion? Give example.
Answer:
During a motion every point in the object traverses a circular path about an axis except the points located on the axis, is called as rotational motion.
Example - Spinning of the earth about its own axis.
Question 6.
What is vibratory motion? Give example.
Answer:
If an object or particle executes a to and fro motion about a fixed point, it is said to be in vibratory motion.
Example - Vibration of a string on a guitar.

Question 7.
What is one dimensional motion? Give example.
Answer:
One dimensional motion is the motion of a particle moving along a straight line. Example - Motion of a train along a straight railway track.
Question 8.
What is two dimensional motion? Give example.
Answer:
If a particle moving along a curved path in a plane, then it is said to be in two dimensional motion.
Example - Motion of a coin on a carrom board.
Question 9.
What is three dimensional motion? Give example.
Answer:
If a particle moving in used three dimensional space, then the particle is said to be in three dimensional motion.
E.g. A bird flying in the sky.

Question 10 .
Write about the properties of components of vectors.
Answer:
If two vectors $\overline{\mathrm{A}}$ and $\overline{\mathrm{B}}$ are equal, then their individual components are also equal. then their individual components are also equal.
$\operatorname{Let} \overline{\mathrm{A}}=\overline{\mathrm{B}}$
then $\mathrm{A}_{\mathrm{x}} \hat{i}+\mathrm{A}_{\mathrm{y}} \hat{j}+\mathrm{A}_{\mathrm{z}} \hat{k}=\mathrm{B}_{\mathrm{x}} \hat{i}+\mathrm{B}_{\mathrm{y}} \hat{j}+\mathrm{B}_{\mathrm{z}} \hat{k}$
i.e. $A_x=B_x, A_y=B_y=A_z=B_z$
Question 11.
Give an example for scalar product of two vectors.
Answer:
The work done by a force $\overrightarrow{\mathrm{F}}$ to move an object through a small displacement $\overrightarrow{\mathrm{dr}}$ then Work done $\mathrm{W}=\overrightarrow{\mathrm{F}} \cdot \overrightarrow{\mathrm{dr}}$ (or) $\mathrm{W}=\mathrm{F} \mathrm{dr} \cos \theta$
Question 12.
Give any three example for vector product of two vectors.
Answer:
1. Torque $\overrightarrow{\mathrm{t}}=\overrightarrow{\mathrm{r}} \times \overrightarrow{\mathrm{F}}$. Where $\mathrm{i}$ is force and $\overrightarrow{\mathrm{F}}$ is force and $\overrightarrow{\mathrm{r}}$ position vector of a particle.
2. Angular momentum $\overrightarrow{\mathrm{L}}=\overrightarrow{\mathrm{r}} \times \overrightarrow{\mathrm{P}}$ where $\overrightarrow{\mathrm{P}}$ is the linear momentum.
3. Linear velocity $\overrightarrow{\mathrm{v}}=\vec{\omega} \times \overrightarrow{\mathrm{r}}$ where $\vec{\omega}$ is angular velocity.

Question 13.
What is position vector?
Answer:
It is vector which denotes the position of a particle at any instant of time, with respect to some
reference frame or coordinate system.
The position $\overrightarrow{\mathrm{r}}$ vector of the particle at a point $P$ is given by $\overrightarrow{\mathrm{r}}=\mathrm{x} \hat{i}+\mathrm{y} \hat{j}+\mathrm{z} \hat{k}$
where $\mathrm{x}, \mathrm{y}$ and $\mathrm{z}$ are components of $\overrightarrow{\mathrm{r}}$.
Question 14.
Write a note an momentum.
Answer:
Momentum of a particle is defined as product of mass with velocity. It is denoted as $\overrightarrow{\mathrm{p}}$ Momentum is also a vector quantity
$
\overrightarrow{\mathrm{r}}=\mathrm{m} \overrightarrow{\mathrm{v}}
$
The direction of momentum is also in the direction of velocity, and the magnitude of momentum is equal to product of mass and speed of the particle.
$
\mathrm{p}=\mathrm{mv}
$

In component form the momentum can be written as $\mathrm{p}_{\mathrm{x}} \hat{i}+\mathrm{p}_{\mathrm{y}} \hat{j}+\mathrm{p}_{\mathrm{z}} \hat{k}=\mathrm{mv}_{\mathrm{x}} \hat{i}+\mathrm{mv}_{\mathrm{y}} \hat{j}+\mathrm{mv}_{\mathrm{z}} \hat{k}$
Here,
$\mathrm{p}_{\mathrm{x}}=\mathrm{x}$ component of momentum and is equal to $\mathrm{mv}_{\mathrm{x}}$
$\mathrm{P}_{\mathrm{x}}=\mathrm{y}$ component of momentum and is equal to $\mathrm{mv}_{\mathrm{y}}$
$P_x=z$ component of momentum and is equal to $\mathrm{mv}_z$
Question 15.
"Displacement vector is basically a position vector". Comment on it.
Answer:
This statement is almost correct only. Because the displacement vector also gives the position of a point just like a position vector. The difference between these two vectors is $p$. The displacement vector gives the position of a point with respect to a point other than origin but position vector gives the position of a point with respect to origin.
Question 16.
Will two dimensional motion with an acceleration only in one dimension?
Answer:
Yes. In oblique projection, the acceleration is acting vertically downward but the object follows a parabolic path.
Question 17.
A foot ball is kicked by a player with certain angle to the horizontal. Is there any point at which velocity is perpendicular to its acceleration.
Answer:
Yes. At its maximum height in the parabolic path vertical velocity is zero but due to horizontal component, velocity acts along horizontally.

Question 18.
Give any two examples for parallelogram law of vectors.
Answer:
- the flight of a bird
- working of a sling.
Question 19.
Why does rubber ball bounce greater heights on hills than in plains?
Answer:
The maximum height attained by the projectile is inversely proportional to acceleration due to gravity. At greater height, acceleration due to gravity will be lesser than plains. So ball can bounce higher in hills than in plains.
Question 20.
Is it possible for body to have variable velocity but constant speed? Give example.
Answer:
Yes, it is possible. In horizontal circular motion the speed of a particle is always constant but due to the variation in direction continuously, the velocity of a particle varies.
Question 21.
What is relative velocity?
Answer:
When two objects are moving with different velocities, then the velocity of one object with respect to another object is called relative velocity of an object with respect to another.
Question 22.
What is average acceleration?
Answer:
The average acceleration is defined as the ratio of change in velocity over the time interval

$\mathrm{a}_{\mathrm{avg}}=\frac{\Delta \overrightarrow{\mathrm{v}}}{\Delta t}$ It is a vector quantity.
Question 23.
Write a note an instantaneous acceleration.
Answer:
Instantaneous acceleration or acceleration of a particle at time ' $t$ ' is given by the ratio of change in velocity over $\Delta \mathrm{t}$, as $\Delta \mathrm{t}$ approaches zero.
Acceleration $\vec{a}=\lim _{\Delta t \rightarrow 0} \frac{\Delta \vec{v}}{\Delta t}=\frac{d \vec{v}}{d t}$
In other words, the acceleration of the particle at an instant $t$ is equal to rate of change of velocity
(1) Acceleration is a vector quantity. Its SI unit is $\mathrm{ms}^{-2}$ and its dimensional formula is $\left[\mathrm{M}^{\circ} \mathrm{L}^1\right.$ $\mathrm{T}^{-2}$ ]
(2) Acceleration is positive if its velocity is increasing, and is negative if the velocity is decreasing. The negative acceleration is called retardation or deceleration.

Question 24 .
Write an acceleration in terms of its component?
(Or)
Show that the acceleration is the second derivative of position vector with respect to time.
Answer:
in terms of components, we can write,
$
\begin{aligned}
& \vec{a}=\frac{d v_x}{d t} \hat{i}+\frac{d v_y}{d t} \hat{j}+\frac{d v_z}{d t} \hat{k}=\frac{d \vec{v}}{d t} \\
& a_x=\frac{d^2 x}{d t^2}, a_y=\frac{d^2 y}{d t^2}, a_z=\frac{d^2 z}{d t^2}
\end{aligned}
$
are the components of instantaneous acceleration. Since each component of velocity is the derivative of the corresponding coordinate, we can express the components $\mathrm{a}_{\mathrm{x}}, \mathrm{a}_{\mathrm{y}}$, and $\mathrm{a}_{\mathrm{z}}$ as $a_x=\frac{d v_x}{d t}, a_y=\frac{d v_y}{d t}, a_z=\frac{d v_z}{d t}$
Then the acceleration vector $\vec{a}$ it self is
$
\vec{a}=\frac{d^2 x}{d t^2} \hat{i}+\frac{d^2 y}{d t^2} \hat{j}+\frac{d^2 z}{d t^2} \hat{k}=\frac{d^2 \vec{r}}{d t^2}
$
Thus acceleration is the second derivative of position vector with respect to time.
Question 25 .
What are the examples of projectile motion?
Answer:

1. An object dropped from window of a moving train.
2. A bullet fired from a rifle.
3. A ball thrown in any direction.
4. A javelin or shot put thrown by an athlete.
5. A jet of water issuing from a hole near the bottom of a water tank.
Question 26.
Explain projectile motion.
Answer:
A projectile moves under the combined effect of two velocities.
- A uniform velocity in the horizontal direction, which will not change provided there is no air resistance.
- A uniformly changing velocity (i.e., increasing or decreasing) in the vertical direction.
There are two types of projectile motion:
- Projectile given an initial velocity in the horizontal direction (horizontal projection)
- Projectile given an initial velocity at an angle to the horizontal (angular projection)

To study the motion of a projectile, let us assume that,
- Air resistance is neglected.
- The effect due to rotation of Earth and curvature of Earth is negligible.
- The acceleration due to gravity is constant in magnitude and direction at all points of the motion of the projectile.
Question 27.
What is time of flight?
Answer:
The time taken for the projectile to complete its trajectory or time taken by the projectile to hit the ground is called time of flight.
Question 28.
Under what condition is the average velocity equal the instantaneous velocity?
Answer:
When the body is moving with uniform velocity.
Question 29.
Draw Position time graph of two objects, A \& B moving along a straight line, when their relative velocity is zero.

Question 30.
Suggest a situation in which an object is accelerated and have constant speed.
Answer:
Uniform Circular Motion.
Question 31.
Two balls of different masses are thrown vertically upward with same initial velocity.
Maximum heights attained by them are $\mathrm{h}_1$ and $\mathrm{h}_2$ respectively what is $\mathrm{h}_1 / \mathrm{h}_2$ ?
Answer:
Same height,
$\therefore \mathrm{h}_1 / \mathrm{h}_2=1$
Question 32 .
A car moving with velocity of $50 \mathrm{kmh}^{-1}$ on a straight road is ahead of a jeep moving with velocity $75 \mathrm{kmh}^{-1}$ would the relative velocity be altered if jeep is ahead of car?
Answer:
No change.

Question 33.
Which of the two - linear velocity or the linear acceleration gives the direction of motion of a body?
Answer:
Linear velocity.
Question 34.
Will the displacement of a particle change on changing the position of origin of the coordinate system?
Answer:
Will not change.
Question 35 .
If the instantaneous velocity of a particle is zero, will its instantaneous acceleration be necessarily zero?
Answer:
No. (highest point of vertical upward motion under gravity).
Question 36.
A projectile is fired with Kinetic energy $1 \mathrm{KJ}$. If the range is maximum, what is its Kinetic energy, at the highest point?
Answer:
Here $\frac{1}{2} \mathrm{mv}^2=1 \mathrm{~kJ}=1000 \mathrm{~J}, \theta=45^{\circ}$
At the highest point, K.E. $=\frac{1}{2} \mathrm{~m}(\mathrm{v} \cos 0)^2=\frac{1}{2} \frac{m v^2}{2}=\frac{1000}{2}=500 \mathrm{~J}$.
Question 37.
Write an example of zero vector.
Answer:
The velocity vectors of a stationary object is a zero vectors.

Question 38.
State the essential condition for the addition of vectors.
Answer:
They must represent the physical quantities of same native.
Question 39.
When is the magnitude of $(\overline{\mathrm{A}}+\overline{\mathrm{B}})$ equal to the magnitude of $(\overline{\mathrm{A}}-\overline{\mathrm{B}})$ ?
Answer:
When $\overline{\mathrm{A}}$ is perpendicular to $\overline{\mathrm{B}}$.
Question 40.
What is the maximum number of component into which a vector can be resolved?
Answer:
Infinite.

Question 41.
A body projected horizontally moves with the same horizontal velocity although it moves under gravity Why?
Answer:
Because horizontal component of gravity is zero along horizontal direction.
Question 42 .
What is the angle between velocity and acceleration at the highest point of a projectile motion?
Answer: $90^{\circ}$.
Question 43.
When does
- height attained by a projectile maximum?
- horizontal range is maximum?
Answer:
- Height is maximum at $\theta=90$
- Range is maximum at $\theta=45$.
Question 44.
What is the angle between velocity vector and acceleration vector in uniform circular motion?
Answer:
$90^{\circ}$.

Question 45.
A particle is in clockwise uniform circular motion the direction of its acceleration is radially inward. If sense of rotation or particle is anticlockwise then what is the direction of its acceleration?
Answer:
Radial in ward.
Question 46.
A train is moving on a straight track with acceleration a. A passenger drops a stone. What is the acceleration of stone with respect to passenger?
Answer:
$\sqrt{a^2+g^2}$ where $\mathrm{g}=$ acceleration due to gravity.
Question 47.
What is the average value of acceleration vector in uniform circular motion over one cycle?
Answer:
Null vector.

Question 48.
Does a vector quantity depends upon frame of reference chosen?
Answer:
No.
Question 49.
What is the angular velocity of the hour hand of a clock?
Answer:
$\omega=\frac{2 \pi}{12}=\frac{\pi}{6} \mathrm{rad} \mathrm{h}^{-1}$
Question 50.
What is the source of centripetal acceleration for earth to go round the sun?
Answer:
Gravitation force of sun.
Question 51 .
What is the angle between $(\vec{A}+\vec{A})$ and $(\vec{A}-\vec{A})$ ?
Answer:
$90^{\circ}$
Short Answer Questions - II (2 Marks)
Question 1.
What are positive and negative acceleration in straight line motion?
Solution:
If speed of an object increases with time, its acceleration is positive. (Acceleration is in the direction of motion) and if speed of an object decreases with time its acceleration is negative (Acceleration is opposite to the direction of motion).

Question 2.
Can a body have zero velocity and still be accelerating? If yes gives any situation.
Solution:
Yes, at the highest point of vertical upward motion under gravity.
Question 3.
The displacement of a body is proportional to $t^3$, where $t$ is time elapsed. What is the nature of acceleration - time graph of the body?
Solution:
As a $\alpha \mathrm{t}^3 \Rightarrow \mathrm{s}=\mathrm{kt}^3$
Velocity, $\mathrm{V}=\frac{d s}{d t}=3 \mathrm{kt}^3$
Acceleration, $\mathrm{a}=\frac{d v}{d t}=3 \mathrm{kt}^3$
i.e., a $\alpha \mathrm{t}$
$\Rightarrow$ motion is uniform, acceleration motion, a - $\mathrm{t}$ graph is straight-line.
Question 4.
Suggest a suitable physical situation for the following graph.

Solution:
A ball thrown up with some initial velocity rebounding from the floor with reduced speed after each hit.
Question 5.
An object is in uniform motion along a straight line, what will be position time graph for the motion of object, if (i) $x_0=$ positive, $v=$ negative is constant.
(i) $\mathrm{x}_0=$ positive, $\mathrm{v}=$ negative is $|\vec{v}|$ constant.
(ii) both $\mathrm{x}_0$ and $\mathrm{v}$ are negative $|\vec{v}|$ is constant.
(iii) $\mathrm{x}_0=$ negative, $\mathrm{v}=$ positive $|\vec{v}|$ is constant.
(iv) both $\mathrm{x}_0$ and $\mathrm{v}$ are positive $|\vec{v}|$ is constant where $\mathrm{x}_0$ is position at $\mathrm{t}=0$.

Solution:

(i)

(ii)

(iii)

(iv)

Question 6.
A cyclist starts from centre $O$ of a circular park of radius $1 \mathrm{~km}$ and moves along the path OPRQO as shown. If he maintains constant speed of $10 \mathrm{~ms}^{-1}$. What is his acceleration at point $\mathrm{R}$ in magnitude \& direction?

Solution:

Centripetal acceleration, $\mathrm{a}_{\mathrm{c}}=\frac{v^2}{r}=\frac{10^2}{1000}=0.1 \mathrm{~m} / \mathrm{s}^2$ along RO.
Question 7.
What will be the effect on horizontal range of a projectile when its initial velocity is doubled keeping angle of projection same?
Solution:
$
\frac{u^2 \sin 2 \theta}{g} \Rightarrow \mathrm{R} \alpha \mathrm{u}^2
$
Range comes four times.

Question 8.
The greatest height to which a man can throw a stone is $\mathrm{h}$. What will be the greatest distance upto which he can throw the stone?
Solution:
Maximum height:
$
\mathrm{H}=\frac{u^2 \sin ^2 \theta}{g} \Rightarrow \mathrm{H}_{\max }=\frac{u^2}{2 g}=\mathrm{h}(\text { at } \theta=90)
$
Maximum range $\mathrm{R}_{\max }=\frac{u^2}{g}=2 \mathrm{~h}$
Question 9.
A person sitting in a train moving at constant velocity throws a ball vertically upwards. How will the ball appear to move to an observer.
- Sitting inside the train
- Standing outside the train
Solution:
- Vertical straight line motion
- Parabolic path.
Question 10.
A gunman always keep his gun slightly tilted above the line of sight while shooting. Why?

Solution:
Because bullet follow Parabolic trajectory under constant downward acceleration.
Question 11.
Is the acceleration of a particle in circular motion not always towards the center. Explain. Solution:
No acceleration is towards the center only in case of uniform circular motion.

Short Answer Questions - III (3 Marks)
Question 1.
Draw
(a) acceleration - time
(b) velocity - time
(c) Position - time graphs representing motion of an object under free fall. Neglect air resistance.
Solution:

$\text { The object falls with uniform acceleration equal to ' } \mathrm{g} \text { ' }$

Question 2.
The velocity time graph for a particle is shown in figure. Draw acceleration time graph from it.

Solutions

Question 3 .
For an object projected upward with a velocity $\mathrm{v}_0$, which comes back to the same point after some time, draw
(i) Acceleration - time graph
(ii) Position - time graph
(iii) Velocity time graph

Question 4.
The acceleration of a particle in $\mathrm{ms}^2$ is given by $\mathrm{a}=3 \mathrm{t}^2+2 \mathrm{t}+2$, where time $\mathrm{t}$ is in second. If the particle starts with a velocity $\mathrm{v}=2 \mathrm{~ms}^{-1}$ at $\mathrm{t}=0$, then find the velocity at the end of $2 \mathrm{~s}$. Solution:
$
\begin{aligned}
& \vec{a}=\frac{d v}{d t}=\left(3 t^2+2 t+2\right) d t \\
& d v=(3 t+2 t+2) d t \\
& \int d v=\int\left(3 t^2+2 t+2\right) d t \\
& v=t^3+t^2+2 t+c \\
& c=2 \mathrm{~m} / \mathrm{s}, v=18 \mathrm{~m} / \mathrm{s} \text { at } t=2 \mathrm{~s}
\end{aligned}
$
Question 5.
At what angle do the two forces $(\mathrm{P}+\mathrm{Q})$ and $(\mathrm{P}-\mathrm{Q})$ act so that the resultant is $\sqrt{3 P^2+Q^2}$ ?
Solution:
Use

$
\begin{aligned}
& \mathrm{R}=\sqrt{3 P^2+Q^2} \\
& \mathrm{R}=\sqrt{3 \mathrm{P}^2+\mathrm{Q}^2} \\
& \mathrm{~A}=\mathrm{P}+\mathrm{Q} \\
& \mathrm{B}=\mathrm{P}-\mathrm{Q} \\
& \text { solve, } \theta=60^{\circ}
\end{aligned}
$
Question 6.
A car moving along a straight highway with speed of $126 \mathrm{~km} \mathrm{~h} 1$ is brought to a stop within a distance of $200 \mathrm{~m}$. What is the retardation of the car (assumed uniform) and how long does it take for the car to stop?
Solution:
Initial velocity of car,
$
\mathrm{u}=126 \mathrm{kmh}^{-1}=126 \times \frac{5}{18} \mathrm{~ms}^{-1}=35 \mathrm{~ms}^{-1}
$
Since, the car finally comes to rest, $\mathrm{v}=0$
Distance covered, $\mathrm{s}=200 \mathrm{~m}, \mathrm{a}=?, \mathrm{t}=$ ?
$
\mathrm{v}^2=\mathrm{u}^2-2 \mathrm{as}
$
or $\mathrm{a}=\frac{v^2-u^2}{2 s}$
substituting the values from eq. (i) in eq . (ii), we get
$
\begin{aligned}
& \mathrm{a}=\frac{0-(35)^2}{2 \times 200}=\frac{0-(35)^2}{2 \times 200} \\
& =-\frac{46}{16} \mathrm{~ms}^{-2}=-3.06 \mathrm{~ms}^{-2}
\end{aligned}
$
Negative sign shows that acceleration in negative which is called retardation, i.e., car is uniformly retarded at $-\mathrm{a}=3.06 \mathrm{~ms}^{-2}$.
To find $t$, let us use the relation
$
\begin{aligned}
& \mathrm{v}=\mathrm{u}+\mathrm{at} \\
& \mathrm{t}=\frac{v-u}{a} \\
& \text { use } \mathrm{a}=-3.06 \mathrm{~ms}^{-2}, \mathrm{v}=0, \mathrm{u}=35 \mathrm{~ms}^{-1} \\
& \therefore \mathrm{t}=\frac{v-u}{a}=\frac{0-35}{-3.06}=11.44 \mathrm{~s} \\
& \therefore \mathrm{t}=11.44 \mathrm{sec}
\end{aligned}
$
Long Answer Questions
Question 1.
Explain the types of motion with example.
Answer:
(a) Linear motion:
An object is said to be in linear motion if it moves in a straight line.
Examples:

- An athlete running on a straight track
- AA particle falling vertically downwards to the Earth.
(b) Circular motion:
Circular motion is defined as a motion described by an object traversing a circular path. Examples:
- The whirling motion of a stone attached to a string.
- The motion of a satellite around the Earth.
- These two circular motions are shown in figure.

(c) Rotational motion:
If any object moves in a rotational motion about an axis, the motion is called 'rotation'. During rotation every point in the object transverses a circular path about an axis, (except the points located on the axis). Examples:
- Rotation of a disc about an axis through its center
- Spinning of the Earth about its own axis.
- These two rotational motions are shown in figure

Examples of
Rotational motion
(d) Vibratory motion:
If an object or particle executes a to-and-fro motion about a fixed point, it is said to be in vibratory motion. This is sometimes also called oscillatory motion.
Examples:
- Vibration of a string on a guitar
- Movement of a swing
- These motions are shown in figure

Other types of motion like elliptical motion and helical motion are also possible.
Question 2.
What are the different types of vectors?
Answer:
1. Equal vectors:
Two vectors $A$ and $B$ are said to be equal when they have equal magnitude and same direction and represent the same physical quantity

Geometrical representation of equal vectors
(a) Collinear vectors:
Col-linear vectors are those which act along the same line. The angle between them can be $0^{\circ}$ or $180^{\circ}$.
(i) Parallel vectors:
If two vectors $\mathrm{A}$ and $\mathrm{B}$ act in the same direction along the same line or on parallel lines, then the angle between them is $0^{\circ}$. Geometrical representation of parallel vectors.

Geometrical representation of parallel vectors
(ii) Anti-parallel vectors:
Two vectors $A$ and $B$ are said to be anti - parallel when they are in opposite directions along
the same line or on parallel lines. Then the angle between them is $180^{\circ}$.

Geometrical representation of anti-parallel vectors.
2. Unit vector:
A vector divided by its magnitude is a unit vector. The unit vector for $\overrightarrow{\mathrm{A}}$ is denoted by $\widehat{A}$. It has a magnitude equal to unity or one.
Since, $\widehat{A}=\frac{\bar{A}}{A}$ we can write $\overrightarrow{\mathrm{A}}=\mathrm{A} \widehat{A}$
Thus, we can say that the unit vector specifies only the direction of the vector quantity.
3. Orthogonal unit vectors:
Let $\hat{i}, \hat{j}$ and $\hat{k}$ be three unit vectors which specify the directions along positive $\mathrm{x}$-axis, positive $\mathrm{y}$-axis and positive z-axis respectively. These three unit vectors are directed perpendicular to each other, the angle between any two of them is $90^{\circ} . \hat{i}, \hat{j}$ and $\hat{k}$ and are examples of orthogonal vectors. Two vectors which are perpendicular to each other are called orthogonal vectors as shown in the figure.

Question 3 .
Explain the concept of relative velocity in one and two dimensional motion.
Answer:
When two objects $A$ and $B$ are moving with different velocities, then the velocity of one object $A$ with respect to another object $B$ is called relative velocity of object $A$ with respect to B.
Case I:
Consider two objects $A$ and $B$ moving with uniform velocities $V_A$ and $V_B$, as shown, along 

straight tracks in the same direction $\overrightarrow{\mathrm{V}}_{\mathrm{A}}, \overrightarrow{\mathrm{V}}_{\mathrm{B}}$ with respect to ground.
The relative velocity of object $A$ with respect to object $B$ is $\vec{V}_{A B}=\vec{V}_A-\vec{V}_B$.
The relative velocity of object $B$ with respect to object $A$ is $\vec{V}_{B A}=\vec{V}_B-\vec{V}_A$ Thus, if two objects are moving in the same direction, the magnitude of relative velocity of one object with respect to another is equal to the difference in magnitude of two velocities.
Case II.
Consider two objects A and B moving with uniform velocities $\vec{V}_A$ and $\vec{V}_B$ along the same straight tracks but opposite in direction.
The relative velocity of an object $A$ with respect to object $B$ is $\overrightarrow{\mathrm{V}}_{\mathrm{AB}}=\overrightarrow{\mathrm{V}}_{\mathrm{A}}-\left(-\overrightarrow{\mathrm{V}}_{\mathrm{B}}\right)=\overrightarrow{\mathrm{V}}_{\mathrm{A}}+\overrightarrow{\mathrm{V}}_{\mathrm{B}}$

The relative velocity of an object $B$ with respect to object $A$ is $\overrightarrow{\mathrm{V}}_{\mathrm{AB}}=-\overrightarrow{\mathrm{V}}_{\mathrm{A}}-\overrightarrow{\mathrm{V}}_{\mathrm{A}}=-\left(\overrightarrow{\mathrm{V}}_{\mathrm{A}}+\overrightarrow{\mathrm{V}}_{\mathrm{B}}\right)$
Thus, if two objects are moving in opposite directions, the magnitude of relative velocity of one object with respect to other is equal to the sum of magnitude of their velocities.
Case III.
Consider the velocities $\vec{v}_A$ and $\vec{v}_B$ at an angle $\theta$ between their directions. The relative velocity of $A$ with respect to $B, \vec{v}_{A B}=\vec{v}_A-\vec{v}_B$
Then, the magnitude and direction of $\overrightarrow{\mathrm{v}}_{\mathrm{AB}}$ is given by $\overrightarrow{\mathrm{v}}_{\mathrm{AB}}=\sqrt{\vec{v}_{\mathrm{A}}^2+\vec{v}_{\mathrm{B}}^2-2 v_{\mathrm{A}} v_{\mathrm{B}} \cos \theta}$ and $\tan \beta=\frac{v_B \sin \theta}{v_A-v_B \cos \theta}$ (Here $\beta$ is angle between $\left(\vec{v}_B\right.$ and $\vec{v}_A$ ) $\overrightarrow{\mathrm{v}}_{\mathrm{A}}-\overrightarrow{\mathrm{v}}_{\mathrm{B}} \cos \theta$
(i) When $\theta=0$, the bodies move along parallel straight lines in the same direction, We have $\vec{v}_{A B}=\left(\vec{v}_A-\vec{v}_B\right)$ in the direction of $\vec{v}_A$. Obviously $\vec{v}_{B A}=\left(\vec{v}_B+\vec{v}_A\right)$ in the direction of $\overrightarrow{\mathrm{v}}_{\mathrm{B}}$
(ii) When $\theta=180^{\circ}$, the bodies move along parallel straight lines in opposite directions, We have $\vec{v}_{A B}=\left(\vec{v}_A+\vec{v}_B\right)$ in the direction of $\vec{v}_A$. Similarly, vBA $=(v B+v A)$ in the direction of $\overrightarrow{\mathrm{v}}_{\mathrm{B}}$

(iii) If the two bodies are moving at right angles to each other, then $0=90^{\circ}$. The magnitude of the relative velocity of $\mathrm{A}$ with respect to $\mathrm{B}=\overrightarrow{\mathrm{v}}_{\mathrm{AB}}=\sqrt{v_{\mathrm{A}}^2+v_{\mathrm{B}}^2}$.
(iv) Consider a person moving horizontally with velocity $\vec{V}_M$. Let rain fall vertically with velocity $\vec{V}_{\mathrm{R}}$. An umbrella is held to avoid the rain. Then the relative velocity of the rain with respect to the person is,

which has magnitude
$
\overrightarrow{\mathrm{V}}_{\mathrm{RM}}=\overrightarrow{\mathrm{V}}_{\mathrm{R}}-\overrightarrow{\mathrm{V}}_{\mathrm{M}}
$
And direction $0=\tan ^{-1}\left(\frac{V_{\mathrm{M}}}{V_{\mathrm{R}}}\right)$ with the vertical as shown in figure.
Question 4.
Shows that the path of horizontal projectile is a parabola and derive an expression for
1. Time of flight
2. Horizontal range
3. resultant relative and any instant
4. speed of the projectile when it hits the ground?
Answer:
Consider a projectile, say a ball, thrown horizontally with an initial velocity $\vec{u}$ from the top of a tower of height $\mathrm{h}$. As the ball moves, it covers a horizontal distance due to its uniform horizontal velocity $\mathrm{u}$, and a vertical downward distance because of constant acceleration due to gravity g. Thus, under the combined effect the ball moves along the path OPA. The motion is in a 2 - dimensional plane. Let the ball take time $\mathrm{t}$ to reach the ground at point $\mathrm{A}$, Then the horizontal distance travelled by the ball is $\mathrm{x}(\mathrm{t})=\mathrm{x}$, and the vertical distance travelled is $\mathrm{y}(\mathrm{t})=$ y.

We can apply the kinematic equations along the $\mathrm{x}$ direction and $\mathrm{y}$ direction separately. Since this is two-dimensional motion, the velocity will have both horizontal component $\mathrm{u}_{\mathrm{x}}$ and vertical component $\mathrm{u}_{\mathrm{y}}$.
Motion along horizontal direction:
The particle has zero acceleration along $\mathrm{x}$ direction. So, the initial velocity ux remains constant throughout the motion. The distance traveled by the projectile at a time $t$ is given by the equation $\mathrm{x}=\mathrm{u}_{\mathrm{x}} t+\frac{1}{2}$ at $^2$. Since $a=0$ along $\mathrm{x}$ direction, we have $\mathrm{x}=\mathrm{u}_{\mathrm{x}} t$
Motion along downward direction:
Here $\mathrm{u}_{\mathrm{y}}=0$ (initial velocity has no downward component), $a=g$ (we choose the $+\mathrm{ve} \mathrm{y}-\mathrm{axis}$ in downward direction), and distance $y$ at time $t$.
From equation, $y=u_x t+\frac{1}{2}$ at $^2$ we get
$
y=\frac{1}{2} \mathrm{at}^2
$

Substituting the value oft from equation (i) in equation (ii) we have
$
\begin{aligned}
& y=\frac{1}{2} g \frac{x^2}{u_z^2}=\left(\frac{g}{2 u_x^2}\right) x^2 \\
& \mathrm{y}=\mathrm{Kx}^2
\end{aligned}
$
where $\mathrm{K}=\frac{g}{2 u_r^2}$ is constant
Equation (iii) is the equation of a parabola. Thus, the path followed by the projectile is a parabola.
1. Time of Flight:
The time taken for the projectile to complete its trajectory or time taken by the projectile to hit the ground is called time of flight. Consider the example of a tower and projectile. Let $\mathrm{h}$ be the height of a tower. Let $\mathrm{T}$ be the time taken by the projectile to hit the ground, after being thrown horizontally from the tower.

Vertical distance covered by the two particles is same in equal intervals.
We know that $\mathrm{s}_{\mathrm{y}}=\mathrm{u}_{\mathrm{y}} \mathrm{t}+\frac{1}{2}$ at ${ }^2$ for vertical motion. Here $. s_{\mathrm{y}}=\mathrm{h}, \mathrm{t}=\mathrm{T}, \mathrm{u}_{\mathrm{y}}=0$ (i.e., no initial
vertical velocity). Then $\mathrm{h}=\frac{1}{2} \mathrm{gt}^2$ or $\mathrm{T}=\sqrt{\frac{2 h}{g}}$ Thus, the time of flight for projectile motion depends on the height of the tower, but is independent of the horizontal velocity of projection. If one ball falls vertically and another ball is projected horizontally with some velocity, both the balls will reach the bottom at the same time. This is illustrated in the Figure
2. Horizontal range:
The horizontal distance covered by the projectile from the foot of the tower to the point where the projectile hits the ground is called horizontal range.
For horizontal motion, we have
$\mathrm{s}_{\mathrm{x}}=\mathrm{u}_{\mathrm{x}} \mathrm{t}+\frac{1}{2} \mathrm{at}^2$
Here, $\mathrm{s}_{\mathrm{x}}=\mathrm{R}$ (range), $\mathrm{u}_{\mathrm{x}}=\mathrm{u}, \mathrm{a}=0$ (no horizontal acceleration) $\mathrm{T}$ is time of flight. Then horizontal range $=\mathrm{uT}$
Since the time of flight $\mathrm{T}=\sqrt{\frac{2 h}{g}}$ we substitute this and we get the horizontal range of the particle as $\mathrm{R}=\mathrm{u} \sqrt{\frac{2 h}{g}}$
The above equation implies that the range $\mathrm{R}$ is directly proportional to the initial velocity $\mathrm{u}$ and inversely proportional to acceleration due to gravity g.
3. Resultant Velocity (Velocity of projectile at any time):
At any instant $\mathrm{t}$, the projectile has velocity components along both $\mathrm{x}$-axis and $\mathrm{y}$-axis. The resultant of these two components gives the velocity of the projectile at that instant $t$, as shown in figure. The velocity component at any $t$ along horizontal (x-axis)
is $\mathrm{V}_{\mathrm{x}}=\mathrm{U}_{\mathrm{x}}+\mathrm{a}_{\mathrm{x}} \mathrm{t}$
Since, $u_x=u, a x=0$, we get
$u_x=u a_x=0$ we get

$
\mathrm{v}_{\mathrm{x}}=\mathrm{u}
$
The component of velocity along vertical direction $(\mathrm{y}-$ axis $)$ is $\mathrm{v}_{\mathrm{y}}=\mathrm{u}_{\mathrm{y}}+\mathrm{a}_{\mathrm{y}} t$ Since, $\mathrm{u}_{\mathrm{y}}=0, \mathrm{a}_{\mathrm{y}}=\mathrm{g}$, we get
$
\mathrm{V}_{\mathrm{y}}=\mathrm{gt}
$
Hence the velocity of the particle at any instant is -
$
\mathrm{v}=\mathrm{u} \hat{i}+\mathrm{g} \hat{j}
$
The speed of the particle at any instant $t$ is given by
$
\begin{aligned}
& \mathrm{v}=\sqrt{v_x^2+v_y^2} \\
& \mathrm{v}=\sqrt{u^2+g^2 t^2}
\end{aligned}
$

4. Speed of the projectile when it hits the ground:
When the projectile hits the ground after initially thrown horizontally from the top of tower of height $\mathrm{h}$, the time of flight is -
$\mathrm{t}=\sqrt{\frac{2 h}{g}}$
The horizontal component velocity of the projectile remains the same i.e $\mathrm{v}_{\mathrm{x}}=\mathrm{u}$.
The vertical component velocity of the projectile at time $\mathrm{T}$ is
$\mathrm{v}=\mathrm{gT}=\mathrm{g} \sqrt{\frac{2 h}{g}}=$
The speed of the particle when it reaches the ground is $\mathrm{v}=\sqrt{u^2+2 g h}$.
Question 5.
Derive the relation between Tangential acceleration and angular acceleration.
Answer:
Consider an object moving along a circle of radius $r$. In a time $\Delta t$, the object travels in an arc distance As as shown in figure. The corresponding angle subtended is $\Delta \theta$
The $\Delta \mathrm{s}$ can be written in terms of $\Delta \theta$ $\Delta \mathrm{s}=\mathrm{r} \Delta \theta \ldots \ldots \ldots(\mathrm{i})$
in a time $\Delta \mathrm{t}$, we have
$
\frac{\Delta s}{\Delta t}=\mathrm{t} \frac{\Delta \theta}{\Delta t}
$
in the limit $\Delta \mathrm{t}-0$, the above equation becomes
$\frac{d s}{d t}=r \omega$
Here $\frac{d s}{d t}$ is linear speed (y) which is tangential to the circle and co is angular speed.
So equation (iii) becomes.

which gives the relation between linear speed and angular speed.
Eq. (iv) is true only for circular motion. In general the relation between linear and angular velocity is given by $\vec{v}=\vec{\omega} \times \vec{r}$
For circular motion eq. (y) reduces to eq. (iv) since $\vec{\omega}$ and $\overrightarrow{\mathrm{r}}$ are perpendicular to each other. Differentiating the eq. (iv) with respect to time, we get (since $r$ is constant) $\frac{d v}{d t}=\frac{r d v}{d t}=\mathrm{r} \alpha$
Here $\frac{d v}{d t}$ Is the tangential acceleration and is denoted as $\mathrm{a}_{\mathrm{t}}=\frac{d \omega}{d t}$ is the angular acceleration $\alpha$. Then eq. (v) becomes
$a_{\mathrm{t}}=\mathrm{r} \alpha$