Elastic Collision — Special Cases

The general formulas

From the elastic collision entry:

$v_1 = \frac{(m_1-m_2)u_1 + 2m_2 u_2}{m_1+m_2}, \quad v_2 = \frac{(m_2-m_1)u_2 + 2m_1 u_1}{m_1+m_2}$

Taking $m_2$ at rest ($u_2 = 0$) for all special cases below.

Case 1: Equal masses ($m_1 = m_2 = m$)

$v_1 = \frac{(m-m)u_1}{2m} = 0$

$v_2 = \frac{2m \cdot u_1}{2m} = u_1$

$m_1$ stops completely. $m_2$ moves off with exactly $m_1$'s initial velocity.

Velocities are exchanged. This is the classic billiard ball result — the cue ball stops dead when it hits another ball of equal mass head-on.

Case 2: Heavy projectile hits light target ($m_1 \gg m_2$)

$v_1 \approx \frac{m_1 u_1}{m_1} = u_1 \quad \text{(barely slows)}$

$v_2 \approx \frac{2m_1 u_1}{m_1} = 2u_1 \quad \text{(moves at twice projectile speed)}$

The heavy body barely changes speed. The light body flies off at twice the projectile's velocity.

Example: A truck colliding elastically with a ping-pong ball. The truck is unaffected; the ball bounces off at high speed.

Case 3: Light projectile hits heavy target ($m_1 \ll m_2$)

$v_1 \approx \frac{-m_2 u_1}{m_2} = -u_1 \quad \text{(bounces back at same speed)}$

$v_2 \approx \frac{2m_1 u_1}{m_2} \approx 0 \quad \text{(barely moves)}$

The light body bounces back with the same speed. The heavy body barely moves.

Example: A ball bouncing off a wall. The wall (Earth) is effectively infinite mass.

Summary table (m₂ at rest)

Why the $m_1 \ll m_2$ case gives $v_2 \approx 2u_1$

Intuitively: imagine the target is a moving wall coming toward you at speed $u_{wall}$. In the wall's frame, the ball approaches at $u_1 + u_{wall}$, bounces back at the same speed, and in the ground frame moves at $u_1 + 2u_{wall}$.

For a stationary wall of infinite mass, this gives $v_2 = u_1 + 2 \times 0 = u_1$... actually let me reclarify:

For case 2 ($m_2$ is the light body): in the limit, the target $m_2$ sees the heavy body $m_1$ as an infinite-mass "wall" at speed $u_1$. In $m_1$'s frame, $m_2$ approaches at $-u_1$, bounces back at $+u_1$. In ground frame: $m_2$ moves at $u_1 + u_1 = 2u_1$. ✓