Question

After a perfectly inelastic collision between two identical balls moving with the same speed in different directions, the speed of the combined mass becomes half of their initial speed. The angle between the velocities of the two balls just before collision is

• A. ${90}^{\circ }$
• B. ${60}^{\circ }$
• C. ${120}^{\circ }$
• D. ${30}^{\circ }$

Answer 1

The correct option is C ${120}^{\circ }$

Let the angle between the velocities of the two balls just before collision be $\theta$.
Since the velocities of both balls are equal, the velocity of each ball will make an angle $\frac{\theta }{2}$ from the $x-$axis as shown in figure.


During collision the forces acting between bodies (Normal reaction) are internal for them. Also the net momentum of system of balls is in $x-$ direction only, as the momentum in $y-$direction cancel each other.

On applying conservation of linear momentum $((F_{\text{ext}}{)}_x=0)$ along $x-$ direction,
$P_i=P_f$
$\Rightarrow mv\cos \frac{\theta }{2}+mv\cos \frac{\theta }{2}=2m\frac{v}{2}$
$\because \frac{v}{2}$ is the final velocity of combined mass.
$\Rightarrow 2v\cos \frac{\theta }{2}=v$
$\Rightarrow \text{cos}\frac{\theta }{2}=\frac{1}{2}$
$\Rightarrow \frac{\theta }{2}={60}^{\circ }$
$\therefore \theta ={120}^{\circ }$