Question

Show that there are two values of time for which a projectile is at the same height. Also show mathematically that the sum of these two times is equal to the time of flight.

Answer 1

From figure, $u_y=u$ $sin\theta$ $a_y=-g$

y-direction : $S_y=u_{y}t+\frac{1}{2}{a}_y{t}^2$

$\therefore$ $h=\left(usin\theta \right)t-\frac{g{t}^2}{2}$

$⟹$ $t^2-(\frac{2u}{g}sin\theta )t+\frac{2h}{g}=0$ ...............(1)

Equation (1) has two roots $t_1$ and $t_2$, thus there are two values of time for which a projectile is at the same height.

Sum of roots of (1) $t_1+t_2=\frac{-b}{a}=\frac{2usin\theta }{g}=T$ (Time of flight)