A body of mass $m$ moving with velocity $v$ collides head on with another body of mass $2\, m$ which is initially at rest. The ratio of $K.E.$ of the colliding body before and after collision will be
$1 : 1$
$2 : 1$
$4 : 1$
$9 : 1$
A curved surface is shown in figure. The portion $BCD$ is free of friction. There are three spherical balls of identical radii and masses. Balls are released from rest one by one from $A$ which is at a slightly greater height than $C$.
With the surface $AB$, ball $1$ has large enough friction to cause rolling down without slipping; ball $2$ has a small friction and ball $3$ has a negligible friction.
$(a)$ For which balls is total mechanical energy conserved ?
$(b)$ Which ball $(s)$ can reach $D$ ?
$(c)$ For ball which do not reach $D$, which of the balls can reach back $A$ ?
A particle of mass $4\, m$ which is at rest explodes into three fragments. Two of the fragments each of mass $m$ are found to move with a speed $v$ each in perpendicular directions. The total energy released in the process will be
A ball $P$ collides with another identical ball $Q$ at rest. For what value of coefficient of restitution $e$, the velocity of ball $Q$ become two times that of ball $P$ after collision
If the potential energy of a gas molecule is
$U = \frac{M}{{{r^6}}} - \frac{N}{{{r^{12}}}}$,
$M$ and $N$ being positive constants, then the potential energy at equilibrium must be
The force $F$ acting on a body moving in a circle of radius $r$ is always perpendicular to the instantaneous velocity $v$. The work done by the force on the body in one complete rotation is :