Conservation of Momentum

When objects interact, they can change each other’s momentum. The total momentum of the objects can nevertheless stay the same: momentum lost by one object is gained by another.

The conservation of momentum principle states that the total momentum of a system remains constant when there is no resultant external force on the system. A system is the set of objects being considered; an external force acts on the system from outside it.

The sums include the momentum of every object in the system, before and after the interaction. An individual object’s momentum can change even though the total remains constant.

Direction and momentum

Momentum is a vector, so its direction matters. For motion along one line, choose one direction as positive. Velocities in that direction are positive; velocities in the opposite direction are negative.

An object’s momentum is $$p = mv$$, where $$m$$ is its mass and $$v$$ is its velocity. Since mass is positive, the sign of the momentum is the same as the sign of the velocity. Add the signed momenta to find the total; do not simply add their magnitudes.

Finding an unknown velocity

A $$2,\mathrm{kg}$$ object moving at $$3,\mathrm{m/s}$$ collides with a $$1,\mathrm{kg}$$ object moving at $$1,\mathrm{m/s}$$ in the opposite direction. After the collision, the $$1,\mathrm{kg}$$ object moves at $$2,\mathrm{m/s}$$ in the original direction of the $$2,\mathrm{kg}$$ object. Assume there is no resultant external force on the two-object system.

Choose the original direction of the $$2,\mathrm{kg}$$ object as positive. The initial velocity of the other object is therefore $$-1,\mathrm{m/s}$$. Let $$v$$ be the final velocity of the $$2,\mathrm{kg}$$ object. Substituting $$p = mv$$ into the conservation equation gives:

Solving gives $$v = 1.5,\mathrm{m/s}$$. The positive result means the $$2,\mathrm{kg}$$ object continues in the chosen positive direction.

The same approach can be used to find an unknown mass: write the total momentum before and after using signed velocities, then solve for the missing quantity.