If the energy and momentum of a ball are E and P, respectively, its velocity will
scale as:
E/P
P/E

The correct answer is A) E/P, because for a particle like a photon the ratio of its energy to momentum equals its velocity.​

Introduction

When a question gives the energy E and momentum P of a moving particle and asks how its
velocity scales, it is really testing the relationship between these three quantities.
For a photon or any massless particle, the fundamental relation is that its velocity equals the ratio of its energy to its momentum,
leading to the answer v = E/P.

Step‑by‑step solution

For a photon, energy and momentum are related to frequency and wavelength as
E = hν and p = h/λ.
Using ν = c/λ, one can show that E/p = c, which is precisely the velocity of the photon.

Thus, the velocity v satisfies
v = E/P,
so the velocity scales directly as E divided by P.

Explanation of every option

Option A: E/P

For photons, the ratio of energy to momentum is equal to the speed of light, a constant velocity,
so the scaling of velocity with E and P is v ∝ E/P.
Therefore, Option A (E/P) is correct.

Option B: P/E

If velocity scaled as P/E, then taking the inverse of the earlier relation would imply
v ∝ 1/c for photons, which contradicts the known constant speed value.
Hence Option B (P/E) is incorrect.

Option C: √(E/P)

A square‑root dependence like √(E/P) does not emerge from the linear energy–momentum relation for photons,
where E is directly proportional to P.
Therefore Option C (√(E/P)) is incorrect.

Option D: E×P

A product E × P would give dimensions and magnitude completely inconsistent with velocity,
since velocity must involve a ratio of energy to momentum, not their product.
Thus Option D (E × P) is also incorrect.