Two bodies are projected from points $(0,0)$ and $(\sqrt{3}-1,0)$ with velocities $10 \ ms^{-1}$ and $v \ ms^{-1}$ respectively,as shown in the figure. The time after which they collide in space is . . . . . . . (in $s$)

$A$ hill is $500 \, m$ high. Supplies are to be sent across the hill,using a cannon that can hurl packets at a speed of $125 \, m/s$ over the hill. The cannon is located at a distance of $800 \, m$ from the foot of the hill and can be moved on the ground at a speed of $2 \, m/s$; so that its distance from the hill can be adjusted. What is the shortest time in which a packet can reach the ground on the other side of the hill? Take $g = 10 \, m/s^2$.

Derive the expressions for the time taken to reach maximum height,the total time of flight,and the maximum height attained by a projectile.

An aircraft executes a horizontal loop with a speed of $150\, m/s$ with its wings banked at an angle of $12^\circ$. The radius of the loop is .......... $km$ $(g = 10\, m/s^2 \text{ and } \tan 12^\circ = 0.2125)$

$A$ particle is projected from a point at an angle $\theta$ with the horizontal direction. At any time $t$, if $p$ is the linear momentum, $y$ is the vertical displacement, and $x$ is the horizontal displacement, which of the following graphs does not represent the variation of kinetic energy $KE$ of the particle?

$A$ projectile is launched with an initial speed of $40 \ m \ s^{-1}$ at an angle of $30^{\circ}$ above the ground. The projectile lands on a hillside $2.0 \ s$ later. The net displacement from where the projectile was launched to where it hits the target is (Take $g = 10 \ m \ s^{-2}$)