Two masses $A\&B$ each of $5 \mathrm{kg}$ are suspended by a light inextensible string passing over a smooth massless pulley such that mass $A$ rest on smooth table & $B$ is held at the position shown. Mass $B$ is now gently lifted up to the pulley and allowed to fall from rest. Determine up to what height(in $\mathrm{m}$) will $A$ rise for the ensuing motion.

 

Three identical balls each of mass $\mathrm{m}=0.5 \mathrm{kg}$ are connected with each other as shown in figure. and rest over a smooth horizontal table. At moment $t=0$, ball $B$ is imparted a horizontal velocity $v_0=9{ \mathrm{ms}}^{-1}$. Calculate velocity of $A$ just before it collides with ball $C$.

Consider the system shown in figure. We know the masses of bodies: $\mathrm{m}=1 \mathrm{kg}, \mathrm{M}=3 \mathrm{kg}$. The distance $AB=BC=\frac{H}{2}$ are covered in equal time intervals by the body of mass $\mathrm{m}$ and at $B$ it undergoes an elastic collision with a stationary body of mass $m_0$ (unknown). Time after this collision when the string in pulley $2$ becomes tight again is $t_0$. $t_0$ is in second. $t_0^2$ is equal to $\frac{1}{x}$. Value of $\frac{x}{10}$ is. (Given : $H=1 \mathrm{m}$ ) (Take: $g=10 \mathrm{m} {\mathrm{s}}^{-2}$)

A wedge of mass $\mathrm{M}=2 \mathrm{m}$ rests on a smooth horizontal plane. A small block of mass $\mathrm{m}$ rests over it at left end $A$ as shown in figure. A sharp impulse is applied on the block, due to which it starts moving to the right with velocity) $v_0=6{ \mathrm{m} \mathrm{s}}^{-1}$. At highest point of its trajectory, the block collides with a particle of same mass $m$ moving vertically downwards with velocity $v=2{ \mathrm{m} \mathrm{s}}^{-1}$ and gets stuck with it. If the combined body lands at the end point $A$ of body of mass $\mathrm{M}$, calculate length $l$ in $\mathrm{cm}$. Neglect friction $\left(g=10{ \mathrm{m} \mathrm{s}}^{-2}\right)$ )

A ball of mass $\mathrm{m}=1 \mathrm{kg}$ is hung vertically by a thread $l=1.50 \mathrm{m}$. Upper end of the thread is attached to the ceiling of a trolley of mass $M=4 \mathrm{kg}$. Initially, trolley is stationary and it free to move along horizontal rails without friction. A shell of mass $\mathrm{m}=1 \mathrm{kg}$ moving horizontally with velocity $v_0=6{ \mathrm{m} \mathrm{s}}^{-1}$ collides with the ball and gets stuck with it. As a result, thread starts to deflected towards right. Calculate its maximum deflection in degrees $\left(°\right)$ with the vertical. $\left(g=10{ \mathrm{ms}}^{-2}\right)$

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.
(i) the coefficient of restitution between the ball and the plates,

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.