Conservation of mechanical energy

Question

Is the mechanical energy after the collision conserved or not conserved? Explain.

Answer 1

Let g be gravitational acceleration.

Both carts start with the same potential energy 2MgH relative to the ground.

When they get to the bottom their potential energy gets converted to kinetic energy.

Let v_A be the velocity of cart A, and v_B be the velocity of cart B.

There is conservation of energy before the crash, so

Mv_A²/2 = 2MgH therefore v_A = 2√gH

2Mv_B²/2 = 2MgH therefore v_B = √2gH

The carts have momenta 2M√gH and 2M√2gH respectively, but in opposite direction.

In a crash, momentum is conserved, so the combined mass of 3M will have momentum

2M√2gH - 2M√gH = 2M√gH(√2-1)

Therefore it will have velocity 2√gH(√2-1)/3.

And it will have kinetic energy

3M(2√gH(√2-1)/3)²/2 = 2MgH(√2-1)²/3

The combination of the two cars after the crash has smaller mechanical energy than the

total mechanical energy of 4MgH before the crash.

Therefore mechanical energy is not conserved; some of it was converted to heat.

This is an example of an "inelastic collision", in which mechanical energy is not conserved.

Collisions where mechanical energy is conserved are called "elastic", and would result

in the two carts bouncing away from each other.