### Centre of Mass, Momentum and Collision

goals
A block of mass 2 kg is attached with a spring of spring constant $4000$ Nm$^{-1}$ and the system is kept on smooth horizontal table. The other end of the spring is attached with a wall. Initially spring is stretched by 5 cm from its natural position and the block is at rest. Now suddenly an impulse of $4$ kg-ms$^{-1}$ is given to the block towards the wall.
A thin rod of length $\ell$ and mass m is suspended horizontally by two vertical strings, A & B, one attached at each end of the rod. The density of the rod is given by $\rho (x)=$${{\rho _ 0}(x/\ell )^ 3}$ where x = 0 corresponds to the position of string A. Give your answer in terms of m, $\ell$ and g.
Two particles, each of mass $m$, are connected by a light inextensible string of length $2l$. Initially they lie on a smooth horizontal table at points $A$ and $B$ distance $l$ apart. the particle at $A$ is projected across the table with velocity $u$. Calculate (in terms of $m$ and $u$) the impulse of tension in the string.
Particle 1 experiences a perfectly elastic collision with a stationary particle 2.
A molecule collides with another, stationary, molecule of the same mass.