Conservation of Momentum

The principle of conservation of momentum states that, in a closed system where no external resultant force acts, the total momentum before an event equals the total momentum after the event.

The equation

total momentum before = total momentum after

m1 v1 + m2 v2 = m1 v1’ + m2 v2’

where m is mass, v is velocity before, and v’ is velocity after.

Collisions

In a collision between two objects, momentum is always conserved (provided no external force acts). There are two types:

Elastic collision: both momentum and kinetic energy are conserved. The objects bounce apart. A good approximation is two billiard balls colliding.

Inelastic collision: momentum is conserved but kinetic energy is not. Some kinetic energy is converted to thermal energy, sound, or deformation. If the objects stick together after the collision, it is perfectly inelastic.

Explosions

Before an explosion, the objects are stationary, so total momentum is zero. After the explosion, the total momentum must still be zero. This means the momenta of the fragments are equal in magnitude and opposite in direction.

Example: a gun (mass 2 kg) fires a bullet (mass 0.01 kg) at 300 m/s.

Before: total momentum = 0 After: momentum of bullet = 0.01 x 300 = 3 kg m/s forward Recoil momentum of gun = 3 kg m/s backward Recoil velocity of gun = 3 / 2 = 1.5 m/s backward

Direction matters

Momentum is a vector quantity. When two objects move in opposite directions, assign one direction as positive and the other as negative. Keep the signs consistent throughout the calculation.