Three objects $A$, $B$ and $C$ are kept in a straight line on a frictionless horizontal surface. These have masses $m, 2 m$ and $m$, respectively. The object $A$ moves towards $B$ with a speed $9 \mathrm{~m} / \mathrm{s}$ and makes an elastic collision with it. Thereafter, $B$ makes completely inelastic collision with $C$. All motions occur on the same straight line. Find the final speed (in $\mathrm{m} / \mathrm{s}$ ) of the object $\mathrm{C}$.

A sphere $P$ of mass $m$ and moving with velocity $v$ undergoes an oblique and perfectly elastic collision with an identical sphere $Q$ initially at rest. The angle $\theta $ between the velocities of the spheres after the collision shall be ............... $^o$

As shown in the figure $a$ body of mass $m$ moving vertically with speed $3\, m/s$ hits a smooth fixed inclined plane and rebounds with a velocity $v_f$ in the horizontal direction. If $\angle$ of inclined is $30^o$, the velocity $v_f$ will be

A ball of mass $m$, moving with a speed $2v_0$, collides inelastically $(e > 0)$ with an identical ball at rest. Show that

$(a)$ For head-on collision, both the balls move forward.

$(b)$ For a general collision, the angle between the two velocities of scattered balls is less than $90^o$.

A mass $m_1$ moves with a great velocity. It strikes another mass $m _2$ at rest in a head on collision. It comes back along its path with low speed, after collision. Then

Two smooth objects with a coefficient of restitution $e,$ collide directly and bounce as shown. Newton's law of restitution gives