A particle of mass $M$ is moving in a horizontal circle ofradius $R$ with uniform speed $v$. When it moves from one point to a diametrically opposite point, its
 

A force acts on a $3\, gm$ particle in such a way that the position of the particle as a function of time is given by $x = 3t -4t^2 + t^3$ , where $x$ is in $meters$ and $t$ is in $seconds$ . The work done during the first $4\, second$ is ................. $\mathrm{mJ}$

A body moving with speed $v$ in space explodes into two piece of masses in the ratio $1 : 3.$ If the smaller piece comes to rest, the speed of the other piece is

A frictionless track $ABCDE$ ends in a circular loop of radius $R$ .A body slides down the track from point $A$ which is at a height $h = 5\, cm$. Maximum value of $R$ for the body to successfully complete the loop is .................. $\mathrm{cm}$

The energy required to accelerate a car from $10 \,m/s$ to $20\, m/s$ is how many times the energy required to accelerate the car from rest to $10\, m/s$

Two ice skaters $A$ and $B$ approach each other at right angles. Skater $A$ has a mass $30\,kg$ and velocity $1\,m/s$ and skater $B$ has a mass $20\,kg$ and velocity $2\,m/s$ . They meet and cling together. Their final velocity of the couple is ............. $\mathrm{m}/ \mathrm{s}$