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 ________.

Answer 1

Step 1. Given data

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.

We have to find the angle between the initial velocities of the two bodies.

Step 2. Formula to be used

Let angle between initial velocities be $\mathrm{\pi }-\mathrm{\theta }$ and the situation is as shown in above figure.

So, conserve the momentum along $\mathrm{x}-$ and $\mathrm{y}-$directions are,

$\mathrm{mv}-\mathrm{mv}·\mathrm{cos\theta }=2\mathrm{m}\frac{\mathrm{v}}{2}\mathrm{cos\alpha }$ for $\mathrm{x}-$ axis.

$\mathrm{mv}s\mathrm{in\theta }=2\mathrm{m}\frac{\mathrm{v}}{2}\mathrm{sin\alpha }$ for $\mathrm{y}-$ axis.

Here, $\mathrm{m}$ is mass and $\mathrm{v}$ is velocity.

Step 3. Find the angle between the initial velocities of the two bodies.

Momentum conservation along $\mathrm{x}-$ axis is,

${\mathrm{mv}}_0\times \mathrm{cos\theta }\times 2=2\mathrm{m}\times \left(\frac{{\mathrm{v}}_0}{2}\right)$

$\mathrm{cos\theta }=\frac{1}{2}$

$\mathrm{\theta }=60°$

Therefore,

$2\mathrm{\theta }=120°$

Hence, the angle between the initial velocities of the two bodies (in degree) is $120°$.