Elastic collisions: they part as fast as they met

5 minAS Level3.3.23.3.3

AI-drafted, independently checked · not yet teacher-reviewed

By the end of this lesson

By the end you can recall that in an elastic collision the relative speed of approach equals the relative speed of separation, and use this with conservation of momentum to find unknown velocities.

Key to the symbols

The main symbols used in this lesson:

Symbol	Meaning
$u_A$	initial velocity of object A
$u_B$	initial velocity of object B
$v_B$	final velocity of object B
$m_A$	mass of object A
$m_B$	mass of object B

The gap closes, the gap opens

Watch the gap between two colliding balls:

before the collision it shrinks at some rate, the relative speed of approach;
after it, the gap grows at some rate, the relative speed of separation.

In an elastic collision those two rates are equal: the objects fly apart exactly as fast as they came together.

A perfectly bouncy ball hitting a wall at $8\ \text{m s}^{-1}$ comes back at $8\ \text{m s}^{-1}$ . Two snooker balls (very nearly elastic) do the same thing relative to each other.

Working out relative speeds

Take the direction from A towards B as positive and give every velocity a sign. Then, for ball A hitting ball B:

$\text{approach} = u_A - u_B \qquad \text{separation} = v_B - v_A$

In words:

moving in the same direction: subtract the speeds;
moving in opposite directions (towards or away from each other): add the speeds.

For an elastic collision: $u_A - u_B = v_B - v_A$ .

B runs away from A at 6.0 m s⁻¹, exactly as fast as A was closing in.

Quick check

Multiple choiceQuestion 1 · Recall

Answer the 2 questions above to continue.