$A$ particle of mass $m$ is moving along the $x$-axis with a speed $v_0$. Suddenly,a part of its mass,$m/3$,breaks off and moves parallel to the $y$-axis with a speed $2v_0$. Find the velocity of the remaining part in terms of unit vectors.

$A$ trolley of mass $1000\, kg$ is moving with a velocity of $50\, km/h$. If a mass of $250\, kg$ is added to it,what will be the new velocity in $km/h$?

$A$ bomb at rest in the air explodes into two fragments. If one fragment moves vertically upwards with a velocity $v_0$,in which direction will the other fragment move?

$A$ man weighing $80 \, kg$ is standing in a trolley weighing $320 \, kg$. The trolley is resting on frictionless horizontal rails. If the man starts walking on the trolley with a speed of $1 \, m/s$ relative to the trolley,then after $4 \, s$ his displacement relative to the ground will be ........ $m$.

$A$ bomb of mass $16 \ kg$ at rest explodes into two fragments of masses $4 \ kg$ and $12 \ kg$. The velocity of the $12 \ kg$ fragment is $4 \ m/s$. What is the kinetic energy of the other fragment in $J$?

$A$ cat with mass $m_c$ is sitting at rest on the left plank of a pair of identical planks. Each plank has mass $m_s$ and they rest on frictionless ice. Suddenly,the cat leaps to the right plank,traveling with a horizontal speed $v_{cg}$ measured with respect to the ground. The instant the cat reaches the right plank,it turns around and leaps back to the left plank. The horizontal component of the cat's speed is again $v_{cg}$ measured with respect to the ground. The final speed of the right plank in terms of the masses of the cat and planks and the cat's leaping speed is (The cat remains on the left plank after its return).