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 .......... $km$ $(g = 10\, m/s^2 \text{ and } \tan 12^\circ = 0.2125)$

$A$ cricketer can throw a ball to a maximum horizontal distance of $100\,m$. With the same speed,how high above the ground can the cricketer throw the same ball (in $,m$)?

An object of mass $m$ is projected with a momentum $p$ at such an angle that its maximum height is $\frac{1}{4}$th of its horizontal range. Its minimum kinetic energy in its path will be

Three point particles $P, Q, R$ move in a circle of radius $r$ with different but constant speeds. They start moving at $t = 0$ from their initial positions as shown in the figure. The angular velocities (in $rad/s$) of $P, Q,$ and $R$ are $5\pi, 2\pi,$ and $3\pi$ respectively, in the same sense. Find the time interval after which they are at the same angular position.

State whether the following statements are true or false:
$(a)$ The time of flight of a projectile depends only on the vertical component of the initial velocity.
$(b)$ The acceleration of a projectile at the maximum height is zero.
$(c)$ The maximum range of a projectile depends on the angle of projection.

$A$ 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?