The range of a projectile is $100 \ m$. Its kinetic energy will be maximum after covering a horizontal distance of: (in $m$)

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 $, m$)?

$A$ particle is moving in the $xy$-plane and crosses the origin at time $t=0$. The equation of motion of the particle is $y=4x^2$. If the velocity of the particle is $\vec{v}=(2\hat{i}+2\hat{j}) \text{ m s}^{-1}$ and acceleration is $\vec{a}=(a\hat{j}) \text{ m s}^{-2}$,then the magnitude of $a$ is

$A$ projectile crosses two walls of equal height $H$ symmetrically as shown. The height of each wall is ........ $m$.

$A$ projectile is thrown upward at an angle $60^{\circ}$ with the horizontal. The speed of the projectile is $20 \text{ m/s}$ when its direction of motion is $45^{\circ}$ with the horizontal. The initial speed of the projectile is . . . . . . in $\text{m/s}$.

$A$ cyclist starts from the centre $O$ of a circular park of radius $1 \text{ km}$,reaches the edge $P$ of the park. Then,the cyclist cycles along the circumference to point $Q$ and returns to the centre along $QO$ as shown in the figure. If the round trip takes $10 \text{ minutes}$,the net displacement and average speed of the cyclist (in metre and kilometre per hour respectively) are: