Question

A projectile can have the same range $R$ far, two angles of projection at a given speed. If $T_1$ and $T_2$ be the times of flight in two cares, then find out relation between $T_1$ , $T_2$ and $R$.

Answer 1

The range of a projectile is given as

$R=\frac{u^2\sin \left(2\theta \right)}{g}..........\left(1\right)$

Let $\theta$ is the angle of projection for first projection and $(90-\theta )$ is the angle of projection for second projection. So,

$T_1=\frac{2u\sin \theta }{g}..........\left(2\right)$

$T_2=\frac{2u\sin (90-\theta )}{g}$

$T_2=\frac{2u\cos \theta }{g}.........\left(3\right)$

Multiplying $T_1\&T_2$

$T_1{T}_2=\frac{4u^2\sin \theta \cos \theta }{g^2}$

$=\frac{2u^2\sin 2\theta }{g^2}(\because \sin 2\theta =2\sin \theta \cos \theta )$

$\frac{2R}{g}=T_1{T}_2$

$R=\frac{T_1{T}_{2}g}{2}$