A flat cart of mass  starts moving to the right due to a constant horizontal force  at . Sand spills on the flat cart from a stationary hopper. The velocity of loading is constant and is equal  kg/s

a)	Initial acceleration is equal to
b)	Acceleration at time  is
c)	Kinetic energy of loaded cart at an instant is equal to work done by force  up to that instant
d)	Momentum of loaded cart at an instant is equal to impulse of force  up to that instant

(a,d)

Let at an instant ‘ ’, velocity of cart be . Due to the sand falling on the cart, it experiences a retarding force, which is equal to  It means net accelerating force on the cart is equal to . But at this instant, mass of the loaded cart is equal to

Therefore, acceleration at time  will be equal to It means option (b) is incorrect.

If in the above equation,  is substituted as zero, then initial acceleration of the cart is

But an initial moment, speed of flat cart is equal to zero. Hence, the initial acceleration becomes . It means, option (a) is correct

If we consider a system of flat cart and falling sand particles, then sand particles exert a retarding force on the cart which is equal to  and the cart exerts an accelerating force on sand particles which is also equal to  It means the only external force or the only resultant force acting on the system is equal to

It means the momentum of the system at any instant  will be equal to impulse of the force . Hence, option (d) is correct

The collision of the sand particles with the cart is perfectly inelastic because just after falling on the cart, sand particles start to move horizontally rightward with the cart while just before coming onto the cart, these sand particles have zero horizontal velocity. It means there is a loss of energy during the collision. Hence, kinetic energy of the loaded cart at an instant will be less than the work done by the force  up to that instant. Hence, option (c) is incorrect