In an inelastic collision,

A ball rolls with a velocity $v$ on a smooth horizontal floor. If $e=$ coefficient of restitution, find the average speed of the ball for to and fro journey.

A ball is projected from the ground at some angle with horizontal. Coefficient of restitution between the ball and the ground is $e.$ Let $a, b$ and $c$ be the ratio of times of flight, horizontal range and maximum height in two successive paths. Find $a, b$ and $c$ in terms of $e.$

$A$ is a fixed point a height $H$ above a perfectly inelastic smooth horizontal plane. A light inextensible string of length $L\left(>H\right)$ has one end attached to $A$ and other to a heavy particle. The particle is held at the level of $A$ with string just taut and released from rest. Find the height of the particle above the plane when it is next instantaneously at rest.

A heavy ball of mass $2M$ moving with a velocity $v_0$ collides elastically head-on with a cradle of three identical balls each of mass $M$ as shown in figure. Determine the velocity of each ball after collision.

In figure, there are $n$ identical spheres of mass $\mathrm{m}$ suspended with wires of equal length. The spheres are almost in contact with each other. Sphere $1$ is pulled aside and released. If sphere $1$ strikes sphere $2$ with velocity $u$ find an expression for velocity $v_n$ of the $n^{th }$ sphere immediately after being struck by the one adjacent to it. The coefficient of restitution for all the impacts is $e.$

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