Two particles of equal mass $m$ are projected from the ground with speeds $v_1$ and $v_2$ at angles ${\theta }_1$ as ${\theta }_2$ shown in the figure.

  

The centre of mass of the two particles,

Two identical rods are joined at one of their ends by a pin. The joint is smooth and rods are free to rotate about the joint. The rods are released in vertical plane on a smooth surface as shown in the figure. The displacement of the joint from its initial position to the final position is (i.e., when the rods lie straight on the ground),

A rail wagon of mass $95\mathrm{ }\mathrm{kg}$ is moving with velocity $10\mathrm{ }\mathrm{m} {\mathrm{s}}^{-1}$ on frictionless rails. A body of mass $5\mathrm{ }\mathrm{kg}$ is thrown into the wagon with velocity $12.5\mathrm{ }\mathrm{m} {\mathrm{s}}^{-1}$ as shown in the figure. The body collides inside with the walls of the wagon and due to internal losses, it comes to rest with respect to the wagon. Find the velocity of the wagon when the body comes to rest w.r.t. the wagon. [Assume that the velocity of the wagon and the object are in the same plane.]

Two men of masses $80 \mathrm{kg}$ and $60 \mathrm{kg}$ are standing on a wooden plank of mass $100 \mathrm{kg}$ that has been placed over a smooth surface. If both the men start moving towards each other with speeds $1 \mathrm{m} {\mathrm{s}}^{-1}$ and $2 \mathrm{m} {\mathrm{s}}^{-1}$, respectively, the velocity of the plank by which it starts moving is,

A block of mass, $m_1=2\mathrm{kg}$ slides on a frictionless table with speed of  $10\mathrm{ }\mathrm{m} {\mathrm{s}}^{-1}$. In front of it, another block of mass $m_2=5\mathrm{kg}$ is moving with speed $3\mathrm{ }\mathrm{m} {\mathrm{s}}^{-1}$ in the same direction and a massless spring of spring constant, $k=1120\mathrm{ }\mathrm{N} {\mathrm{m}}^{-1}$ is attached on the backside of $m_2$ as shown. The maximum compression of the spring in $\mathrm{cm}$ when the blocks collide is,

A man is standing on a cart of mass which is double the mass of the man. Initially, the cart is at rest on the smooth ground. Now, the man jumps with relative velocity $v$, horizontally towards right with respect to the cart. The work done by the man during the process of jumping is,

A block of mass $2m$ is kept on a smooth circular track of mass $m$ which is kept on a smooth horizontal surface. The circular track is given a horizontal velocity $\sqrt{2gR}$, towards left and is released. The maximum height reached by $2m$ will be,