Explanation

(d) Figure shows the particles each moving with same speed $v$ but in different directions. Consider any two particles having angle $\theta$ between directions of their velocities
[image:13_image27.emf]
Then, $\overrightarrow{v_{rel}} = \overrightarrow{v_{B}} - \overrightarrow{v_{A}}$
$i.e.,\ v_{rel} = \sqrt{v^{2} + v^{2} - 2vv\cos\theta}$
$\Rightarrow v_{rel} = \sqrt{2v^{2}(1 - \cos\theta)} = 2v\sin{(\theta/2)}$
So averaging $v_{rel}$ over all pairs
${\overline{v}}_{rel} = \frac{\int_{0}^{2\pi}{v_{rel}d\theta}}{\int_{0}^{2\pi}{d\theta}} = \frac{\int_{0}^{2\pi}{2v\sin{(\theta/2)}}}{\int_{0}^{2\pi}{d\theta}} = \frac{2v \times 2{\lbrack - \cos{(\theta/2)}\rbrack}_{0}^{2\pi}}{2\pi}$
$\Rightarrow {\overline{v}}_{rel} = (4v/\pi) > v$ [as $4/\pi > 1$]