Question

A particle moving on a circular path with constant speed $V$, light is emitted from a projector at the centre of the circular path as shown in figure. The shadow of the particle is formed on the wall. The velocity of shadow up the wall is

• A. $V{\sec }^2\varphi$
• B. $V{\cos }^2\varphi$
• C. $V{\cos }^{}\varphi$
• D. $V{\sec }^{}\varphi$

Answer 1

The correct option is A $V{\sec }^2\varphi$

Let us assume r is the radius of the circular path and u be the velocity of the shadow

The position of the shadow on wall is given by

$x=r\tan \varphi$

Derivating w.r.t t on both sides of the equation,

$\frac{d}{dt}\left(x\right)=\frac{d}{dt}\left(r\tan \varphi \right)$

$u=r{\sec }^2\varphi \frac{d\varphi }{dt}$

Substituting $\omega =\frac{d\varphi }{dt}$ in the above equation,

$u=r\omega {\sec }^2\varphi$

$u=V{\sec }^2\varphi$

The velocity of the shadow up the wall is $V{\sec }^2\varphi$