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

An object of mass $2m$ is projected with a speed of $100 \ m/s$ at an angle $\theta = \sin^{-1}(3/5)$ to the horizontal. At the highest point,the object breaks into two pieces of same mass $m$ and the first one comes to rest. The distance between the point of projection and the point of landing of the second piece (in metres) is $(g = 10 \ m/s^2)$.

An isolated rail car of mass $M$ is moving along a straight,frictionless track at an initial speed $v_0$. The car is passing under a bridge when a crate filled with $N$ bowling balls,each of mass $m$,is dropped from the bridge into the bed of the rail car. The crate splits open and the bowling balls bounce around inside the rail car,but none of them fall out. What is the average speed of the rail car $+$ bowling balls system some time after the collision?

$A$ person of mass $M=90 \ kg$ standing on a smooth horizontal plane of ice throws a body of mass $m=10 \ kg$ horizontally on the same surface. If the distance between the person and the body after $10 \ s$ is $10 \ m$,then the kinetic energy $(KE)$ of the person (in $J$) is: (in $J$)

$A$ bomb of mass $3.0 \, kg$ explodes in air into two pieces of masses $2.0 \, kg$ and $1.0 \, kg$. The smaller mass travels at a speed of $80 \, m/s$. The total energy imparted to the two fragments is ............. $kJ$.

$A$ particle of mass $5m$ at rest suddenly breaks on its own into three fragments. Two fragments of mass $m$ each move along mutually perpendicular directions with speed $v$ each. The energy released during the process is