Question

Pankaj and Sudhir are playing with two different balls of masses $m$ and $2m$, respectively. If Pankaj throws his ball vertically up and Sudhir at an angle $\theta$, both the balls stay in our view for the same period. Find the ratio of the heights attained by the two balls.

Answer 1

Time of flight for the ball thrown by

Pankaj, $T_1=\frac{2u_1}{g}$

Time of flight for the ball thrown by Sudhir,

$T_2=\frac{2u_2\sin ({90}^o-\theta )}{8}=\frac{2u_2\cos \theta }{g}$

According to $T_1=T_2\Rightarrow \frac{2u_1}{g}=\frac{2u_2\cos \theta }{g}$

$\Rightarrow$ to $T_1=T_2\Rightarrow \frac{2u_1}{g}=\frac{2u_2\cos \theta }{g}$

$\Rightarrow u_1=u_2\cos \theta$

Height of the ball thrown by pankaj ,$H_1=\frac{u_1^2}{2g}$

Height of the ball thrown by Sudhir,

$H_2=\frac{u_2^2{\sin }^2({90}^o-\theta )}{2g}=\frac{u_2^2{\cos }^2\theta }{2g}$

$\therefore \frac{H_1}{H_2}=\frac{u_1^2/2g}{u_2^2{\cos }^2\theta /2g}=1$ [as $u_1=u_2\cos \theta ]$