H2 Physics 6 - Collisions and Impulse: Head-On Elastic Collision

Step 1 - Set up the problem

A 2 kg ball at 6 m/s right collides head-on with a 3 kg ball at 2 m/s left.

Step 1 - Set up the problem

The collision is perfectly elastic. Find both velocities after impact.

Step 1 - Set up the problem

Take rightward as positive.

Step 2 - Apply conservation of momentum

Conservation of momentum: total before = total after.

Step 2 - Apply conservation of momentum

2(6) + 3(-2) = 2v_1 + 3v_2.

Step 2 - Apply conservation of momentum

12 - 6 = 2v_1 + 3v_2, giving 6 = 2v_1 + 3v_2. This is equation one.

Step 3 - Apply the elastic collision condition

For elastic collisions: relative speed of approach = relative speed of separation.

Step 3 - Apply the elastic collision condition

Approach speed = 6 - (-2) = 8 m/s.

Step 3 - Apply the elastic collision condition

Separation speed: v_2 - v_1 = 8. This is equation two.

Step 4 - Solve the simultaneous equations

From (2): v_2 = v_1 + 8.

Step 4 - Solve the simultaneous equations

Substitute into (1): 6 = 2v_1 + 3(v_1 + 8).

Step 4 - Solve the simultaneous equations

6 = 5v_1 + 24.

Step 4 - Solve the simultaneous equations

5v_1 = -18, so v_1 = -3.6 m/s.

Step 5 - Interpret and verify

Ball 1 bounces back at 3.6 m/s left. Ball 2 moves at 4.4 m/s right.

Step 5 - Interpret and verify

Check momentum: 2(-3.6) + 3(4.4) = -7.2 + 13.2 = 6. Matches initial momentum.

Step 5 - Interpret and verify

Both conservation laws satisfied. The lighter ball reverses - typical for a lighter ball hitting a heavier one.