# Induced velocity $w$ at a point $z=z_{1}$ along the lifting line can be calculated using the formula $w\left(z_{1}\right)=-\frac{1}{4 \pi} \int_{-s}^{S} \frac{d \Gamma}{d z} \frac{1}{z-z_{1}} d z$ Given $\frac{\Gamma^{2}}{\Gamma_{o}^{2}}+\frac{z^{2}}{s^{2}}=1,$ where $z, \Gamma_{o}$ and sare given in figure below. For theabove semi-elliptic distribution of circulation, $\Gamma,$ the downwash velocity at any point $Z_{1}$, for symmetric flight can be obtained as, $w\left(z_{1}\right)=\frac{\Gamma_{o}}{4 \pi s}\left[\pi+z_{1} I\right],$ where $I=z_{1} \int_{-s}^{S} \frac{d z}{\sqrt{s^{2}-z^{2}}\left(z-z_{1}\right)}$ Which of the following options is correct if the induced drag is $D_{i}$ (given $\int_{-s}^{s} \sqrt{1-\frac{z^{2}}{s^{2}}} d z=\frac{\pi s}{2}$ ) $(\mathrm{A}) I=0$ and $D_{i}=\frac{8 \rho \Gamma_{o}^{2}}{\pi}$ $(\mathrm{B}) I=1$ and $D_{i}=\frac{8 \rho \Gamma_{o}^{2}}{\pi}$ $(\mathrm{C}) I=0$ and $D_{i}=\frac{\pi \rho \Gamma_{o}^{2}}{8}$ $(\mathrm{D}) I=I$ and $D_{i}=\frac{\pi \rho \Gamma_{o}^{2}}{8}$

## Question ID - 155910 | SaraNextGen Top Answer Induced velocity $w$ at a point $z=z_{1}$ along the lifting line can be calculated using the formula $w\left(z_{1}\right)=-\frac{1}{4 \pi} \int_{-s}^{S} \frac{d \Gamma}{d z} \frac{1}{z-z_{1}} d z$ Given $\frac{\Gamma^{2}}{\Gamma_{o}^{2}}+\frac{z^{2}}{s^{2}}=1,$ where $z, \Gamma_{o}$ and sare given in figure below. For theabove semi-elliptic distribution of circulation, $\Gamma,$ the downwash velocity at any point $Z_{1}$, for symmetric flight can be obtained as, $w\left(z_{1}\right)=\frac{\Gamma_{o}}{4 \pi s}\left[\pi+z_{1} I\right],$ where $I=z_{1} \int_{-s}^{S} \frac{d z}{\sqrt{s^{2}-z^{2}}\left(z-z_{1}\right)}$ Which of the following options is correct if the induced drag is $D_{i}$ (given $\int_{-s}^{s} \sqrt{1-\frac{z^{2}}{s^{2}}} d z=\frac{\pi s}{2}$ ) $(\mathrm{A}) I=0$ and $D_{i}=\frac{8 \rho \Gamma_{o}^{2}}{\pi}$ $(\mathrm{B}) I=1$ and $D_{i}=\frac{8 \rho \Gamma_{o}^{2}}{\pi}$ $(\mathrm{C}) I=0$ and $D_{i}=\frac{\pi \rho \Gamma_{o}^{2}}{8}$ $(\mathrm{D}) I=I$ and $D_{i}=\frac{\pi \rho \Gamma_{o}^{2}}{8}$

$(\mathrm{C}) I=0$ and $D_{i}=\frac{\pi \rho \Gamma_{o}^{2}}{8}$