Two persons of mass $m_1$ and $m_2$ are standing at the two ends $A$ and $B$ respectively,of a trolley of mass $M$ as shown. When both the persons jump simultaneously with $u_{rel}$ with respect to the trolley,then the velocity of the trolley is

$A$ bullet of mass $40 \,g$ is fired from a gun of mass $10 \,kg$. If the velocity of the bullet is $400 \,m/s$,then the recoil velocity of the gun will be ..........

$A$ projectile is moving at $20 \, m/s$ at its highest point,where it breaks into two equal parts due to an internal explosion. One part moves vertically up at $30 \, m/s$. Then the other part will move at ............. $m/s$.

$A$ ball of mass $10 \, kg$ moving with a velocity $10 \sqrt{3} \, m/s$ along the $X$-axis,hits another ball of mass $20 \, kg$ which is at rest. After the collision,the first ball comes to rest and the second one disintegrates into two equal pieces. One of the pieces starts moving along the $Y$-axis at a speed of $10 \, m/s$. The second piece starts moving at a speed of $20 \, m/s$ at an angle $\theta$ (in degrees) with respect to the $X$-axis. The configuration of pieces after the collision is shown in the figure. The value of $\theta$ to the nearest integer is:

In an explosion,a body breaks up into two pieces of unequal masses. In this process:

$A$ bomb of $12 \ kg$ divides into two parts whose ratio of masses is $1 : 3$. If the kinetic energy of the smaller part is $216 \ J$,then the momentum of the bigger part in $kg \cdot m/s$ will be: