$A$ particle is performing a uniform circular motion along a circle of radius $R$. In half the period of revolution,its displacement and distance covered are respectively:

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 $, m$)?

$A$ ball of mass $160 \, g$ is thrown up at an angle of $60^{\circ}$ to the horizontal at a speed of $10 \, m/s$. The angular momentum of the ball at the highest point of the trajectory with respect to the point from which the ball is thrown is nearly $\left(g=10 \, m/s^{2}\right)$ (in $kg \cdot m^{2}/s$).

$A$ horizontal plane supports a stationary vertical cylinder of radius $R = 1 \ m$ and a disc $A$ attached to the cylinder by a horizontal thread $AB$ of length $l_0 = 2 \ m$ (seen in figure,top view). An initial velocity $v_0 = 1 \ m/s$ is imparted to the disc as shown in the figure. How many seconds will it move along the plane until it strikes against the cylinder? (All surfaces are assumed to be smooth.)

$A$ particle is acted upon by a force of constant magnitude which is always perpendicular to the velocity of the particle. The motion of the particle takes place in a plane. It follows that:

The ratio between the maximum range and the square of the time of flight in projectile motion is: