Question

A small particle elastically strikes a fixed smooth sphere of $4cm$ radius, with a speed of $5m/s$. The impact parameter for the collision is $2\sqrt{3}cm$.Calculate the angle of deflection of the particle.

• A. ${60}^{\circ }$
• B. ${30}^{\circ }$
• C. ${45}^{\circ }$
• D. ${75}^{\circ }$

Answer 1

The correct option is A ${60}^{\circ }$

$\begin{array}{c}\text{Given, impact parameter}\\ \text{and the collision}\\ \text{is elastic.}\\ \Rightarrow \theta ={60}^{\circ }cm\\ \Rightarrow sin\theta =\frac{2\sqrt{3}}{4}=\frac{\sqrt{3}}{2}\\ \Rightarrow \theta =60\end{array}$

Since, the sphere is fixed, then momentum of the small particle is changed along the $y^{\prime }$ axis only

$e=1=\frac{v\cos \varphi }{5/2}\Rightarrow v\cos \varphi =\frac{5}{2}$

Along axis $x^{\prime }$, there is no force so no change in the speed.

$\begin{array}{cc}5\sin 60& =\mathrm{vsin}\varphi \\ v\sin \varphi & =\frac{5\sqrt{3}}{2}\end{array}$

Upon equating (1) \& (2) we get.

$\begin{array}{cc}\tan \varphi & =\sqrt{3}\\ \varphi & ={60}^{\circ }\\ & \end{array}$

$\therefore \Delta \varphi =180-(\theta +\varphi )={60}^{\circ }$