$A$ projectile fired at $30^{\circ}$ to the ground is observed to be at the same height at time $t_1 = 3 \, s$ and $t_2 = 5 \, s$ after projection,during its flight. The speed of projection of the projectile is $......... \, m \, s^{-1}$ (Given $g = 10 \, m \, s^{-2}$).

$A$ shot is fired from a point at a distance of $200 \ m$ from the foot of a tower $100 \ m$ high,such that it just passes over it. The direction of the shot with respect to the horizontal is (in $^{\circ}$)

$A$ ball is hit by a batsman at an angle of $37^o$ with an initial velocity $u = 15 \ m/s$ as shown in the figure. The man standing at $P$ is at a distance of $9 \ m$ from the point $B$ where the ball would strike the ground. What is the minimum velocity at which the man should run to catch the ball before it hits the ground? Assume the height of the man is negligible. ........ $m/s$

$A$ particle moves in a plane along an elliptic path given by $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$. At point $(0, b)$,the $x$-component of velocity is $u$. The $y$-component of acceleration at this point is

The ratio of minimum kinetic energies of two projectiles of same mass is $4: 1$ and the ratio of maximum heights attained by them is $4: 1$. Then the ratio of their ranges is . . . . . . (in $: 1$)

$A$ particle has an initial velocity of $(3\hat i + 4\hat j) \; ms^{-1}$ and an acceleration of $(0.4\hat i + 0.3\hat j) \; ms^{-2}$. Its speed after $10 \; s$ is: