A ball is thrown from the location $\left(x_0, y_0\right)=(0,0)$ of a horizontal playground with an initial speed $v_0$ at an angle $\theta_0$ from the $+x$-direction. The ball is to be hit by a stone, which is thrown at the same time from the location $\left(x_1, y_1\right)=(L, 0)$. The stone is thrown at an angle $\left(180-\theta_1\right)$ from the $+x$-direction with a suitable initial speed. For a fixed $v_0$, when $\left(\theta_0, \theta_1\right)=\left(45^{\circ}, 45^{\circ}\right)$, the stone hits the ball after time $T_1$, and when $\left(\theta_0, \theta_1\right)=\left(60^{\circ}, 30^{\circ}\right)$, it hits the ball after time $T_2$. In such a case, $\left(T_1 / T_2\right)^2$ is. . . . .

The maximum range of a gun on horizontal terrain is $16 \,km$. If $g = \;10m/{s^2}$. What must be the muzzle velocity of the shell ......... $m/s$

A particle reaches its highest point when it has covered exactly one half of its horizontal range. The corresponding point on the displacement time graph is characterised by

At the height $80 \,m$, an aeroplane is moving with $150\, m/s$. A bomb is dropped from it so as to hit a target. At what distance from the target should the bomb be dropped ......... $m$.

A ball rolls off the top of a stairway with horizontal velocity $\mathrm{u}$. The steps are $0.1 \mathrm{~m}$ high and $0.1 \mathrm{~m}$ wide. The minimum velocity $\mathrm{u}$ with which that ball just hits the step $5$ of the stairway will be $\sqrt{\mathrm{x}} \mathrm{ms}^{-1}$ where $\mathrm{x}=$___________ [use $\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^2$ ].

A plane is flying horizontally at $98\, m/s$ and releases an object which reaches the  ground in $10 \sec$. The angle made by object while hitting the ground is   ......... $^o$