Two bodies of the same mass are moving with the same speed, but in different directions in a plane. They have a completely inelastic collision and move together thereafter with a final speed which is half of their initial speed. The angle between the initial velocities of the two bodies (in degree) is ________.

Solution

Physics sir · 3 steps

Momentum is conserved as a vector, so add the two equal velocities and match the size of the result.

1. Conserve momentum. Let each body have mass m and speed v, with angle θ between their velocities. After the perfectly inelastic collision they move together as mass 2m with speed v/2.

   mv⃗₁ + mv⃗₂ = 2m V⃗ ⇒ |v⃗₁ + v⃗₂| = 2 × v/2 = v

2. Size of the sum. For two vectors of equal size v at angle θ, |v⃗₁ + v⃗₂|² = v² + v² + 2v²cos θ. Setting this equal to v² gives 2v²cos θ = -v², so cos θ = -1/2.
3. So. cos θ = -1/2 means θ = 120°. The answer is 120.

The common slip: 60. Using the shortcut 2v cos θ/2 = v and then reporting the half-angle as the answer gives 60, not 120. Here θ/2 = 60°, so θ is twice that.

The idea to keep. In a 2D collision, conserve momentum as a vector. Two equal vectors at angle θ add up to 2v cos θ/2. Remember the half-angle when you use that form.

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