perfectly inelastic collision
A perfectly inelastic collision is the ultimate stick-together crash: the two objects lock into one lump and move off with a single shared velocity. A dart burying itself in a board, a train car coupling to a wagon, a snowball splatting onto a moving sled — after the hit, there is only one object moving.
Because they end up moving together, the final velocity is fixed by momentum conservation alone: m1 v1 + m2 v2 = (m1 + m2) v_final. This is the collision that loses the MOST kinetic energy allowed while still conserving momentum — but note it does not lose ALL of it. If the combined lump is still moving (because the total momentum was not zero), it must keep the kinetic energy that goes with that motion. Only when the total momentum is zero (equal and opposite) does everything stop and all the kinetic energy convert to other forms.
This special case is a gift for problem-solving, because 'they stick together' removes an unknown: instead of two final velocities you have just one. It underlies the ballistic pendulum (a bullet embedding in a block to measure its speed) and is the standard first model for car-crash and coupling problems.
A 10 g bullet at 400 m/s buries itself in a 2.0 kg block at rest. They move off together at v = (0.010 times 400) / (0.010 + 2.0) = 4 / 2.01 = about 1.99 m/s. Kinetic energy crashes from 1/2 times 0.010 times 400^2 = 800 J to about 4 J — over 99% became heat inside the block. Momentum, 4 kg m/s, is unchanged.
Stick-together collision: one final velocity, and almost all the kinetic energy gone.
Perfectly inelastic does not mean everything stops. The lump stops only if the total momentum was zero to begin with; otherwise it must keep moving to carry that momentum, retaining the kinetic energy that goes with it.