Consider a collection of large number of particles, each...

Consider a collection of large number of particles, each moving with a speed $v$. The direction of velocity is randomly distributed in the collection. The magnitude of the relative velocity between a pair of particles averaged over all the pairs in the collection (A) $v$ (B) $\frac{2 v}{\pi}$ (C) $\frac{\pi v}{4}$ (D) $\frac{4 v}{\pi}$

Consider a collection of large number of particles, each moving with a speed $v$. The direction of velocity is randomly distributed in the collection. The magnitude of the relative velocity between a pair of particles averaged over all the pairs in the collection
(A) $v$
(B) $\frac{2 v}{\pi}$
(C) $\frac{\pi v}{4}$
(D) $\frac{4 v}{\pi}$

Key Concepts

Relative Velocity

Relative velocity is the speed at which one particle appears to be moving from the point of view of another particle. In physics, it is obtained by subtracting one particle's velocity vector from the other’s, and is crucial in analyzing interactions such as collisions, where the outcome depends on the relative motion of the particles involved.

Vector Subtraction and Trigonometry

Vector subtraction involves calculating the difference between two velocity vectors, which depends not only on the magnitudes of the velocities but also on the angle between them. In the case of fixed-magnitude velocities, trigonometric relationships such as the sine function are employed to determine the magnitude of the difference, leading to expressions that involve sin(?/2) when the angle between the two vectors is ?.

Random (Isotropic) Distribution of Directions

When the directions of particle velocities are randomly distributed, the system is said to be isotropic, meaning that all directions are equally probable. This property allows for the use of uniform probability distributions over the relevant angular range when calculating average quantities, such as the mean relative speed between pairs of particles.

Averaging over Angular Distributions

Averaging over all pairs of particles requires integrating the expression for relative velocity over the distribution of the angle between velocity vectors. Since the angle between randomly oriented vectors has a well-defined probability distribution, the integration using appropriate limits yields the average relative speed. This mathematical averaging process is essential in statistical mechanics and kinetic theory to derive macroscopic properties from microscopic dynamics.

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