Elastic Collisions in 1D & 2D

When no kinetic energy is lost we get two conservation laws for two unknowns, and the outcome of a head-on collision is fixed completely. Billiard balls, Newton’s cradle and gas molecules all obey the same two formulas.

Head-on elastic collision#

Masses $m_1, m_2$ move along one line with velocities $u_1, u_2$ before and $v_1, v_2$ after. Write both conservation laws with each mass’s terms collected on one side:

Divide the second by the first (the factors of difference-of-squares cancel nicely):

This is linear, so combining it with momentum conservation is easy:

Three cases worth knowing

• Equal masses: $v_1 = u_2, \ v_2 = u_1$. The bodies simply swap velocities. A moving ball hitting an identical one at rest stops dead and the other carries on (Newton’s cradle).
• Very heavy target at rest ($m_2 \gg m_1$): $v_1 \approx -u_1, \ v_2 \approx 0$. A ball bounces back off a wall with its speed unchanged.
• Very heavy projectile ($m_1 \gg m_2$, target at rest): $v_1 \approx u_1, \ v_2 \approx 2u_1$. A bowling ball barely notices a ping-pong ball, which flies off at twice its speed.

Collisions in two dimensions#

If the balls don’t meet head-on, the push during contact acts along the line joining their centres. Momentum is conserved as a vector, so we get one equation for each component, plus energy if the collision is elastic. That is three equations for four unknowns (two speeds and two angles), so one more piece of information, such as one outgoing angle, is needed.