A ball moving vertically downward with a speed of $10 \mathrm{m} {\mathrm{s}}^{-1}$ collides with a platform. The platform moves with a velocity of $5 \mathrm{m} {\mathrm{s}}^{-1}$ in downward direction. If $e=0.8,$ find the speed (in $\mathrm{m} {\mathrm{s}}^{-1}$ ) of the ball just after collision.

A body of mass $1 \mathrm{kg}$ moving with velocity $1 \mathrm{m}/\mathrm{s}$ makes an elastic one dimensional collision with an identical stationary body. They are in contact for brief time $1 \mathrm{s}.$ Their force of interaction increases from zero to $F_0$ linearly in time $0.5 \mathrm{s}$ and decreases linearly to zero in further time $0.5 \mathrm{s}$ as shown in figure. Find the magnitude of force $F_0$ in newton.

A $70 \mathrm{g}$ ball $B$ dropped from a height $h_0=9 \mathrm{m}$reaches a height $h_2=0.25 \mathrm{m}$ after bouncing twice from identical $210 \mathrm{g}$plates. Plate A rests directly on hard ground, while plate $C$ rests on a foam -rubber mat. Determine.
(ii) the height $h_1$ of the ball's first bounce.

A stream of glass beads, each with a mass of $15$ gram, comes out of a horizontal tube at a rate of $100$ per second. The beads fall a distance of $5 \mathrm{m}$ to a balance pan and bounce back to their original height. How much mass (in $\mathrm{kg}$) must be placed in the other pan of the balance to keep the pointer at zero?

A $5 \mathrm{kg}$ sphere is connected to a fixed point $O$ by an inextensible string of length $5 \mathrm{m}$. The sphere is resting on a horizontal surface at a distance $4 \mathrm{m}$ from $O$. The sphere is given a vertical velocity $v_0$, and it moves freely till it reaches point $P$, when string becomes taut. Find the maximum allowable velocity ${\mathrm{v}}_0$ (in m/s), if impulse of tension in string is not to exceed $6 \mathrm{N} \mathrm{s}$.

A coordinate axis system taking $x$ -axis as horizontal smooth floor is shown in figure. Two small balls of masses $m$ and $3 \mathrm{m}$ attached with a string are released from some heights on $y$ -axis as shown in figure. The balls may collide head on or obliquely. After a certain time mass $\mathrm{m}$ is at $(9 \mathrm{cm}, 20 \mathrm{cm})$ while mass $3 \mathrm{m}$ is $25 \mathrm{cm}$ above the $x$ axis and the string is taut. The balls always remain in $x-y$ plane. Find the length of string (in $\mathrm{cm}$)

A boy is playing carom board. Strikers is moving with velocity $u(=10 \mathrm{m}/\mathrm{s})$ and queen $Q$ is at rest. See angle $\theta \left(=37°\right)$ in the figure. Mass of $S$ and $Q$ are $m$ and $2 \mathrm{m}.$ Friction is absent and coefficient of resistution is $e.$ If after collision $Q$ moves along $SQ$ direction and $S$ moves along perpendicular to $SQ.$ Find the value of coefficient of restitution.

Mass of a particle and a hemisphere are $m$ and $3 \mathrm{m}/2$ respectively. Particle hits the hemisphere (at rest on smooth surface) with a horizontal velocity $v_0$ and rebounds in vertical direction after collision (neglect friction), if the value of $\theta$ (in degree) (collision is elastic):