Newtonian mechanics

Inelastic collisions

An inelastic collision does not conserve mechanical kinetic energy; an isolated system still conserves total momentum.

Some initial kinetic energy becomes internal energy, heat, sound, or deformation. If the objects stick together, the collision is perfectly inelastic.

Solving a collision

Choose a system containing both objects, set a positive direction, and apply momentum conservation. For a one-dimensional perfectly inelastic collision, $V=(m_1u_1+m_2u_2)/(m_1+m_2)$. Transformed energy is not destroyed.

Quick check

The kinetic-energy decrease in a perfectly inelastic collision can be calculated from the initial and final states, but that energy is not destroyed: it becomes internal energy, sound, and deformation. For masses $m_1,m_2$ with initial velocities $u_1,u_2$ along one line, the lost kinetic energy is $Delta K_{lost}= frac12mu(u_1-u_2)^2$, where $mu=m_1m_2/(m_1+m_2)$ is the reduced mass. The expression shows that initial relative speed controls the loss.

Use signed velocities for opposite directions before substituting; the final common velocity is not an arithmetic mean unless the masses are equal. Check the answer by multiplying the final velocity by the combined mass to recover the initial total momentum. A negative result means the joined bodies move in the chosen negative direction.

For example, two equal-mass carts moving in opposite directions at equal speeds have zero total initial momentum; if they stick, they remain at rest. This does not violate the law: zero momentum is conserved while the initial kinetic energy becomes other forms. The example shows why one object’s velocity alone cannot predict the joined motion.