A ball with a speed of 9 m/s collides with another identical ball at rest. After the collision, the direction of each ball makes an angle of 30° with the original direction. The ratio of velocities of the balls after collision is x : y, where x is __________.

Solution

Physics sir · 4 steps

Look at the momentum sideways: there was none before, so the two sideways parts must cancel after.

1. Set up axes. Take the x-axis along the original direction of motion and the y-axis perpendicular to it. Before the collision, all the momentum is along x and the y-momentum is zero.
2. Perpendicular momentum. After the collision, the balls move off at 30° on opposite sides of the x-axis with speeds v₁ and v₂. Both have the same mass m, so y-momentum conservation gives the equation below.

   m v₁ sin 30° - m v₂ sin 30° = 0 ⇒ v₁ = v₂

3. Check along x. Along x, 2v cos 30° = 9 m s⁻¹, so each ball moves at v = 3√3 ≈ 5.2 m s⁻¹. You do not need this speed for the ratio, and the initial 9 m/s does not change it.
4. So. The speeds are equal, so the ratio is 1 : 1 and x = 1.

The idea to keep. In a 2D collision, write momentum conservation along both axes separately. The axis with zero initial momentum often gives the answer straight away.

Answer from the NTA final answer key. Solution by PrepWiser.