A particle of mass m is moving along the x-axis with initial velocity uî. It collides elastically with a particle of mass 10 m at rest and then moves with half its initial kinetic energy (see figure). If sin θ ₁ = √n sin θ ₂ then value of n is ________.

Solution

Physics sir · 4 steps

Use energy to get both final speeds, then balance momentum perpendicular to the x-axis.

1. Speed of m. Its kinetic energy becomes half, so 1/2mv₁² = 1/2(1/2mu²). That gives v₁ = u/√2, not u/2.
2. Speed of 10m. The collision is elastic, so the other half of the kinetic energy goes to 10m: 1/2(10m)v₂² = 1/4mu². So v₂² = u²/20 and v₂ = u/√20.
3. Perpendicular momentum. Before the collision there is no momentum along y. After it, the y-components of the two momenta must cancel.

   m u/√2 sin θ₁ = 10m u/√20 sin θ₂ ⇒ sin θ₁ = 10√2/√20 sin θ₂ = √10 sin θ₂

4. So. Comparing with sin θ₁ = √n sin θ₂ gives n = 10.

The common slip: 30. Taking half the kinetic energy as half the speed (v₁ = u/2) makes v₂ = u√(3/40) and gives 30, not 10.

The idea to keep. In a 2D elastic collision, energy gives you the final speeds and momentum perpendicular to the original motion links the angles. Half the kinetic energy means speed divided by √2.

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