Two particles of equal mass are projected simultaneously with speeds $20 \mathrm{m} {\mathrm{s}}^{-1}$ and $10\sqrt{3} \mathrm{m} {\mathrm{s}}^{-1}$ as shown in figure. Find the maximum height reached by the centre of mass of the particles.

 

A man of mass $50 \mathrm{kg}$ is standing on a $100 \mathrm{kg}$ plank kept on a frictionless horizontal floor. Initially both are at rest. If the man starts walking on the plank with speed $6 \mathrm{m} {\mathrm{s}}^{-1}$ towards right relative to the plank, then amount of muscular energy spent by the man is

Two smooth prisms of similar right-triangular sections are arranged on a smooth horizontal plane, as shown in figure. The lower prism has a mass $\eta$ times the upper prism. The prisms are held in an initial position as shown and are then released. As the upper prism touches the horizontal plane, the distance moved by the lower prism is

A loaded spring gun initially at rest on a horizontal frictionless surface fires a marble at angle of elevation $\theta$. The mass of the gun is $M$, the mass of the marble is $m$, and the muzzle velocity of the marble is $v_0$. The final velocity of the gun is

Two blocks of mass $2\mathrm{kg}$ and $5\mathrm{kg}$ are given speed as shown in the figure. System is lying on a frictionless surface and the blocks are connected by a massless spring if spring constant $35{\mathrm{Nm}}^{-1}$. Find the maximum compression in the spring.

A body of mass $M$ is moving on a smooth surface with speed $v_0$. Another body of same mass $M$ attached to a spring is kept on the same surface as shown. The moving mass makes an impact with the spring of spring constant $k$ and starts compressing it. What is the maximum compression in the spring?

A uniform rod of mass $M$ and length $L$ is standing vertically along the $y-$axis on a smooth horizontal surface as shown. A small disturbance causes the lower end to slip on the smooth horizontal surface and rod starts falling. The path followed by centre of mass of the rod during the fall is