When is momentum conserved but kinetic energy is not

A closed system experiences no net external force, so the vector sum of its momenta remains constant according to Newton's third law. This statement holds regardless of the microscopic details of the interaction. However, kinetic energy depends on the squares of speeds, and any process that diverts part of that energy into non‑mechanical forms will reduce the mechanical kinetic energy while still preserving total momentum.

Inelastic collision example

Consider two identical 1 kg carts on a frictionless track. Cart A moves rightward at 4 m/s, while Cart B is initially at rest. They stick together after a perfectly inelastic collision. Momentum before collision is \(p_{i}=1\times4+1\times0=4\,\text{kg·m/s}\). The combined mass after collision is 2 kg, giving a common velocity $v_f = p_i / (2) = 2\,\text{m/s}$. The final kinetic energy is $K_f = \tfrac12 (2)(2^2)=4\,\text{J}$, half the initial 8 J, showing kinetic energy is not conserved even though momentum is.

Typical situations where kinetic energy is not conserved:

• Perfectly inelastic collisions (objects stick together)
• Partially inelastic collisions (some deformation)
• Collisions that generate heat or sound
• Explosions where internal chemical energy converts to kinetic energy

How to verify momentum conservation in a collision:

1. 1Write the momentum of each object before the event $p_i = m_1 v_{1i}+m_2 v_{2i}$.
2. 2Write the momentum after the event $p_f = m_1 v_{1f}+m_2 v_{2f}$.
3. 3Set $p_i = p_f$ and solve for any unknown final speed.

Numerical comparison of the example before and after the collision:

Energy conservation never fails; the missing kinetic energy is transferred to internal degrees of freedom, such as microscopic deformation, heat, or sound waves. Because internal forces are equal and opposite, they cancel in the momentum balance, leaving the total momentum unchanged. Recognizing the distinction between momentum (a vector conserved by symmetry) and kinetic energy (a scalar that can be converted) is essential for solving exam problems on collisions.