Question

If at an instant the velocity of a projectile be $u$ and its inclination to the horizontal be $\theta$, at what time interval after that instant will the particle be moving at right angles to its former direction.

Answer 1

Let the velocity of the particle at $t$ = 0 be $u$ at direction $\Theta$ to the horizontal.

Velocity at (t=0) , $u_1$ = $ucos\Theta \overrightarrow{i}+usin\Theta \overrightarrow{j}$

Velocity at time t, $u_2$ = $ucos\Theta \overrightarrow{i}+(usin\Theta -gt)\overrightarrow{j}$

For $u_1$ and $u_2$ to be perpendicular ,

$u_1.u_2=0$

$u^{2}co{s}^2\Theta +u^{2}si{n}^2\Theta -ugtsin\Theta =0$

$t$ = $\frac{u}{gsin\Theta }$

Therefore , the particle moves at right angles to its former direction at $t$= $\frac{u}{gsin\Theta }$