Find the velocity of the centre of mass of two identical particles moving with velocities `v_(1)` and `v_(2)`.

Show that in absence of any external force the centre of mass of two moving particles moves with uniform velocity.

Find the expression for the kinetic energy of an object of mass m, moving with velocity v.

A particle travels half of the distance between A and B with a velocity v_(0) . During half of the time required to travel the remaining distance the particle moves with a velocity v_(1) and the rest with a velocity v_(2) . Find the average velocity of the particle.

A particle starting from a point A and moving with a positive constant acceleration along a straight line reaches another point B is time T. Suppose that the initial velocity of the particle is u gt0 and P is the midpoint of the line AB. If the velocity of the particle at point P is v_(1) and if the velocity at time (T)/(2) is v_(2) , then -

Two particles of masses m_1 and m_2 move on the same plane. If their relative velocity is u, and the velocity of their centre of mass is v, show that the total kinetic energy is 1/2(m_1+m_2)v^2+1/2 ((m_1m_2)/(m_1+m_2))u^2 .

Two bodies having masses m_(1) and m_(2) are dropped from heights h_(1) and h_(2) respectively. They reach the ground after times t_(1) and t_(2) and strike the ground with velocities v_(1) and v_(2) respectively. Choose the correct relations from the following:

A body of mass m moving with velocity V along the X-axis, collides with another mass M moving with velocity v along the Y-axis. The masses coalesce after collision. Find the velocity and the direction of motion of the combined mass.