An aeroplane executes a horizontal loop at a speed of $720 \text{ km/h}$ with its wings banked at $45^{\circ}$. What is the radius of the loop (in $\text{ km}$)? Take $g = 10 \text{ m/s}^2$.

$A$ sportsman runs around a circular track of radius $r$ such that he traverses the path $ABAB$. The distance travelled and displacement,respectively,are

The coordinates of a moving particle at any time $t$ are given by $x = \alpha t^3$ and $y = \beta t^3$,where $\alpha$ and $\beta$ are constants. The speed of the particle at time $t$ is given by:

The hour hand of a clock is $6\,cm$ long. The magnitude of the displacement of the tip of the hour hand between $1:00\,PM$ and $5:00\,PM$ is:

$A$ particle is projected vertically upwards from $O$ with velocity $v$ and a second particle is projected at the same instant from $P$ (at a height $h$ above $O$) with velocity $v$ at an angle of projection $\theta$. The time when the distance between them is minimum is

There are two points $P$ and $Q$ on a projectile path with velocities $v_P$ and $v_Q$ respectively,such that $v_P$ is perpendicular to $v_Q$. If $\alpha$ is the angle that $v_P$ makes with the horizontal at point $P$,find the correct option.