A moving particle of mass $m$, makes a head on elastic collision with another particle of mass $2m$, which is initially at rest. The percentage loss in energy of the colliding particle on collision, is close to
(1) $33 \%$
(2) $67 \%$
(3) $90 \%$
(4) $10 \%$

A simple pendulum, made of a string of length $l$ and a bob of mass $m$, is released from a small angle $\theta _ { 0 }$. It strikes a block of mass $M$, kept on horizontal surface at its lowest point of oscillations, elastically. It bounces back and goes up to an angle $\theta _ { 1 }$. Then $M$ is given by:
(1) $m \left( \frac { \theta _ { 0 } - \theta _ { 1 } } { \theta _ { 0 } + \theta _ { 1 } } \right)$
(2) $m \left( \frac { \theta _ { 0 } + \theta _ { 1 } } { \theta _ { 0 } - \theta _ { 1 } } \right)$
(3) $\frac { m } { 2 } \left( \frac { \theta _ { 0 } + \theta _ { 1 } } { \theta _ { 0 } - \theta _ { 1 } } \right)$
(4) $\frac { m } { 2 } \left( \frac { \theta _ { 0 } - \theta _ { 1 } } { \theta _ { 0 } + \theta _ { 1 } } \right)$

Two particles of masses $M$ and 2 M are moving with speeds of $10 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and $5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, as shown in the figure. They collide at the origin and after that they move along the indicated directions with speeds $v _ { 1 }$ and $v _ { 2 }$, respectively. The values of $v _ { 1 }$ and $v _ { 2 }$ are, nearly
(1) $6.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and $3.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$
(2) $3.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and $12.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }$
(3) $13.02 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and $19.7 \mathrm {~m} \mathrm {~s} ^ { - 1 }$
(4) $3.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and $6.3 \mathrm {~m} \mathrm {~s} ^ { - 1 }$

A body A of mass $m = 0.1 \mathrm {~kg}$ has an initial velocity of $3 \hat { \mathrm { i } } \mathrm { m s } ^ { - 1 }$. It collides elastically with another body B of the same mass which has an initial velocity of $5 \hat { \mathrm { j } } \mathrm { m s } ^ { - 1 }$. After the collision, A moves with a velocity $\vec { v } = 4 ( \hat { \mathrm { i } } + \hat { \mathrm { j } } ) \mathrm { m s } ^ { - 1 }$. The energy of B after the collision is written as $\frac { x } { 10 } \mathrm {~J}$. The value of $x$ is $\_\_\_\_$.