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A particle of mass m moving with speed u collides elastically with a sphere of radius r

Question :- A particle of mass m moving with speed u collides elastically with a sphere of radius r and same mass at rest, at an impact parameter b. Find
(a) Angle between their final velocities and
(b) Magnitude of their final velocities.

Solution :

Let the line of impact is along x-axis, so

$\displaystyle \overrightarrow{{{u}_{1}}}=ucos\theta \hat{i}+usin\theta \widehat{j}$

$\displaystyle \overrightarrow{{{u}_{2}}}=0$

As the collision is perfectly elastic and masses of both the bodies are equal, thus their velocity vectors will get exchanged along the line of impact after the collision. Thus final velocities after collision :-

$\displaystyle \overrightarrow{{{v}_{1}}}=0\hat{i}+usin\theta \widehat{j}=usin\theta \widehat{j}$

$\displaystyle \overrightarrow{{{v}_{2}}}=ucos\theta \hat{i}+0\widehat{j}=ucos\theta \hat{i}$

Also if we focus on the particles, they will experience force (normal reaction) along the line of impact and there is no force perpendicular to it, i.e., along the common tangent. Thus their velocities does not change along common tangent.

Now from geometry,

$\displaystyle sin\theta =\frac{b}{r}$