Question

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 ________.

Hint:

Conserve linear momentum: the vector sum of two equal momenta of magnitude mv at angle \(\theta\) has magnitude \(2mv\cos(\theta/2)\).

Answer 1

Correct Answer: 120

Let each body have mass m and speed v, with angle \(\theta\) between the velocities. After the perfectly inelastic collision the combined mass 2m moves with speed v/2.
Momentum conservation: \(|\vec p_1+\vec p_2|=2m\cdot\frac{v}{2}=mv\).
\(|\vec p_1+\vec p_2|=\sqrt{(mv)^2+(mv)^2+2(mv)^2\cos\theta}=2mv\cos\frac{\theta}{2}\).
So \(2mv\cos\frac{\theta}{2}=mv\Rightarrow\cos\frac{\theta}{2}=\frac12\Rightarrow\frac{\theta}{2}=60^\circ\).
\(\theta=120^\circ\).
Answer: 120