$A$ ball is released from rest from point $P$ of a smooth semi-spherical vessel as shown in the figure. The ratio of the centripetal force and normal reaction on the ball at point $Q$ is $A$,while the angular position of point $Q$ is $\alpha$ with respect to point $P$. Which of the following graphs represents the correct relation between $A$ and $\alpha$ when the ball goes from $Q$ to $R$?

Two balls are thrown as shown in the figure and they reach the ground in the same time. What is the ratio of the maximum heights attained by them?

$A$ particle starts from the origin at $t=0$ with a velocity $5 \hat{i} \text{ m/s}$ and moves in the $x-y$ plane under the action of a force which produces a constant acceleration of $(3 \hat{i} + 2 \hat{j}) \text{ m/s}^2$. If the $x$-coordinate of the particle at that instant is $84 \text{ m}$,then the speed of the particle at this time is $\sqrt{\alpha} \text{ m/s}$. The value of $\alpha$ is . . . . . . .

$A$ particle of mass $m$ is executing uniform circular motion on a path of radius $r$. If $p$ is the magnitude of its linear momentum,then the radial force acting on the particle is:

$A$ body of mass $100 \, g$ is projected at an angle of $30^\circ$ with the horizontal with a velocity of $20 \, m \, s^{-1}$. What is the change in its momentum at the maximum height in $kg \, m \, s^{-1}$?

Two particles $A$ and $B$ are moving on two concentric circles of radii $R_1$ and $R_2$ with equal angular speed $\omega$. At $t = 0$,their positions and directions of motion are shown in the figure. The relative velocity $\overrightarrow{V_A} - \overrightarrow{V_B}$ at $t = \frac{\pi}{2\omega}$ is given by: