A body of mass 2 kg moving with a speed of $4\mathrm{~m~s}^{-1}$ makes an elastic collision with another body at rest and continues to move in the original direction but with one fourth of its initial speed. The speed of the two body centre of mass is $\frac{x}{10}$ m/s. Find the value of $x$.

A rod of mass $M$ and length $L$ is lying on a horizontal frictionless surface. A particle of mass $m$ travelling along the surface hits at one end of the rod with a velocity $u$ in a direction perpendicular to the rod. The collision is completely elastic. After collision, particle comes to rest. The ratio of masses ( $\frac { m } { M }$ ) is $\frac { 1 } { x }$. The value of $x$ will be

A ball of mass 10 kg moving with a velocity $10 \sqrt { 3 } \mathrm {~m} \mathrm {~s} ^ { - 1 }$ along the $x$-axis, hits another ball of mass 20 kg which is at rest. After the collision, first ball comes to rest while the second ball disintegrates into two equal pieces. One piece starts moving along $y$-axis with a speed of $10 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The second piece starts moving at an angle of $30 ^ { \circ }$ with respect to the $x$-axis. The velocity of the ball moving at $30 ^ { \circ }$ with $x$-axis is $x \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The configuration of pieces after the collision is shown in the figure below. The value of $x$ to the nearest integer is [Figure]

A body of mass 8 kg and another of mass 2 kg are moving with equal kinetic energy. The ratio of their respective momenta will be
(1) $1 : 1$
(2) $2 : 1$
(3) $1 : 4$
(4) $4 : 1$

A body of mass $M$ at rest explodes into three pieces, in the ratio of masses $1 : 1 : 2$. Two smaller pieces fly off perpendicular to each other with velocities of $30 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and $40 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ respectively. The velocity of the third piece will be
(1) $35 \mathrm {~m} \mathrm {~s} ^ { - 1 }$
(2) $50 \mathrm {~m} \mathrm {~s} ^ { - 1 }$
(3) $25 \mathrm {~m} \mathrm {~s} ^ { - 1 }$
(4) $15 \mathrm {~m} \mathrm {~s} ^ { - 1 }$

A man of 60 kg is running on the road and suddenly jumps into a stationary trolly car of mass 120 kg. Then the trolly car starts moving with velocity $2\mathrm{~m\,s^{-1}}$. The velocity of the running man was $\_\_\_\_$ $\mathrm{m\,s^{-1}}$, when he jumps into the car.

Two bodies are having kinetic energies in the ratio $16:9$. If they have same linear momentum, the ratio of their masses respectively is:
(1) $3:4$
(2) $9:16$
(3) $16:9$
(4) $4:3$

A simple pendulum of length 1 m has a wooden bob of mass 1 kg. It is struck by a bullet of mass $10 ^ { - 2 } \mathrm {~kg}$ moving with a speed of $2 \times 10 ^ { 2 } \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The bullet gets embedded into the bob. The height to which the bob rises before swinging back is. (use $g = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 }$)
(1) 0.30 m
(2) 0.20 m
(3) 0.35 m
(4) 0.40 m

A body of mass 5 kg moving with a uniform speed $3 \sqrt { 2 } \mathrm {~m} \mathrm {~s} ^ { - 1 }$ in $X - Y$ plane along the line $y = x + 4$. The angular momentum of the particle about the origin will be $\_\_\_\_$ $\mathrm { kg } \mathrm { m } ^ { 2 } \mathrm {~s} ^ { - 1 }$.