$A$ strap is passing over a wheel of radius $30\, cm.$ During the time the wheel moving with an initial constant velocity of $2\, rev/s$ comes to rest,the strap covers a distance of $25\, m.$ The deceleration of the wheel in $rad/s^2$ is

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 $, m$)?

$A$ projectile of mass $m$ is projected from the ground with a speed of $50 \, m/s$ at an angle of $53^{\circ}$ with the horizontal. It breaks up into two equal parts at the highest point of the trajectory. One particle comes to rest immediately after the explosion. The ratio of the radii of curvatures of the moving particle just before and just after the explosion is:

Two cars of masses $m_{1}$ and $m_{2}$ are moving in circles of radii $r_{1}$ and $r_{2}$ respectively. Their speeds are such that they make complete circles in the same time $t$. The ratio of their centripetal force is

$A$ truck starts from rest and accelerates uniformly at $2.0 \; m s^{-2}$. At $t = 10 \; s$,a stone is dropped by a person standing on the top of the truck ($6 \; m$ high from the ground). What are the $(a)$ velocity,and $(b)$ acceleration of the stone at $t = 11 \; s$? (Neglect air resistance.)

Two stones of masses $m$ and $2\,m$ are whirled in horizontal circles,the heavier one in a radius $\frac{r}{2}$ and the lighter one in radius $r$. The tangential speed of the lighter stone is $n$ times that of the heavier stone when they experience the same centripetal force. The value of $n$ is