$A$ body at rest breaks into three fragments. Two fragments of equal mass fly off perpendicular to each other with a speed of $30 \; m/s$. The mass of the third fragment is three times that of the other two fragments. What will be the direction and magnitude of the velocity of the third fragment immediately after the explosion?

Two masses $A$ and $B$ of mass $M$ and $2M$ respectively are connected by a compressed ideal spring. The system is placed on a horizontal frictionless table and given a velocity $u \hat{k}$ in the $z$-direction as shown in the figure. The spring is then released. In the subsequent motion,the line from $B$ to $A$ always points along the $\hat{i}$ unit vector. At some instant of time,mass $B$ has an $x$-component of velocity as $V_x \hat{i}$. The velocity $\vec{V}_A$ of mass $A$ at that instant is:

$A$ body initially at rest breaks up into two pieces of masses $2 M$ and $3 M$ respectively,together having a total kinetic energy $E$. The piece of mass $2 M$,after breaking up,has a kinetic energy:

$A$ 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.

$A$ man of mass $m$ is standing on a cart of mass $2m$. Initially, the cart is at rest. The man jumps horizontally with a relative velocity $u$ with respect to the cart. The work done by the internal forces of the man during the process of jumping will be:

$A$ 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 ........ $m$.