The position coordinates of a projectile projected from the ground on a certain planet (with no atmosphere) are given by $y = (4t - 2t^2) \text{ m}$ and $x = (3t) \text{ m}$,where $t$ is in seconds and the point of projection is taken as the origin. The angle of projection of the projectile with the vertical is ......... (in $^{\circ}$)

Two stones are projected so as to reach the same horizontal range on a horizontal surface. The maximum height reached by one exceeds the other by an amount equal to half the sum of the heights attained by them. Then,the angle of projection of the stone which attains the smaller height is $........$

$A$ particle is moving eastwards with a speed of $6 \, m/s$. After $6 \, s$,the particle is found to be moving with the same speed in a direction $60^{\circ}$ north of east. The magnitude of average acceleration in this interval of time is ....... $m/s^2$.

The coordinates of a moving particle at a time $t$ are given by $x = 5 \sin 10t$ and $y = 5 \cos 10t$. The speed of the particle is:

$A$ body is projected with velocity $u$ making an angle $\alpha$ with the horizontal. Its velocity when it is perpendicular to the initial velocity vector $u$ is

In a circus,a performer throws an apple towards a hoop held at $45 \, m$ height by another performer standing on a high platform (see figure). The thrower aims for the hoop and throws the apple with a speed of $24 \, m/s$. At the exact moment that the thrower releases the apple,the other performer drops the hoop. The hoop falls straight down. At what height above the ground does the apple go through the hoop (in $, m$)?