$A$ body of mass $M$ at rest explodes into three pieces,two of which of mass $M/4$ each are thrown off in perpendicular directions with velocities of $3 \, m/s$ and $4 \, m/s$ respectively. The third piece will be thrown off with a velocity of .......... $m/s$

$A$ rifleman,who together with his rifle has a mass of $100\,kg$,stands on a smooth surface and fires $10$ shots horizontally. Each bullet has a mass of $10\,g$ and a muzzle velocity of $800\,ms^{-1}$. The velocity which the rifleman attains after firing $10$ shots is $..........\,ms^{-1}$.

$A$ man is standing at the centre of a frictionless pond of ice. How can he get himself to the shore?

$A$ person of mass $80 \ kg$ is standing on a trolley of mass $320 \ kg$. The trolley is at rest on a frictionless horizontal rail. If the person walks on the trolley at a speed of $1 \ m/s$,what will be their displacement relative to the ground after $4 \ s$?

$A$ projectile is fired with velocity $u$ at an angle $\theta$ with the horizontal. At the highest point of its trajectory,it splits into three segments of masses $m, m$,and $2m$. The first part falls vertically downward with zero initial velocity,and the second part returns via the same path to the point of projection. The velocity of the third part of mass $2m$ just after the explosion will be:

Three particles of masses $10 \ g, 20 \ g$ and $40 \ g$ are moving with velocities $10 \hat{i}, 10 \hat{j}$ and $10 \hat{k} \ m/s$ respectively. If due to some mutual interaction,the first particle comes to rest and the velocity of the second particle becomes $(3 \hat{i} + 4 \hat{j}) \ m/s$,then the velocity of the third particle is: