Question

A small particle travelling with a velocity v collides elastically with a spherical body of equal mass and of radius r initially kept at rest. The centre of this spherical body is located a distance rho(< r) away from the direction of motion of the particle. Find the final velocities of the two particles.
Figure

Answer 1

The final velocities of the particles are explained below.

Explanation:

Let the particle mass as well as the spherical body be m.

See now the figure

The velocity of particles (v) has two components:

v cos α It's normal in the field

v sin α The sphere is tangential

After the impact, speeds are exchanged.

So, the spherical body will have a velocity v cos α and in this direction the particle will have no velocity variable.

Collision occurred because the v cos α portion is normal

Yet, particle vs sin α's tangential velocity will not be affected.

Now , The velocity of the sphere = v cos α =$v \sqrt{\frac{\left(r^{2}-\rho^{2}\right)}{r}}$

Velocity of particle = v sin α = $\frac{\mathrm{v} \rho}{r}$