If the particles were moving with a speed much less than c, the magnitude of the momentum of the second particle would be twice that of the first. However, what is the ratio of the magnitudes of momentum for these relativistic particles?

The moment is a very useful concept, since it is one of the quantities that is conserved during shocks and explosions, for which it had to be redefined to be consistent with special relativity,

p = m v / √[1+ (v/c)² ]

for the case of speeds much lower than the speed of light this expression is close to

p = m v

In this exercise they indicate that the moment of the second particle is twice the moment of the first, when their velocities are small

p₂ = 2 p₁

p₂/p₁ = 2

in consecuense

m v₂ = 2 m v₁

v₂ = 2 v₁

consider particles of equal mass.

By the time their speeds increase they enter the relativistic regime

p₂ = mv₂ /√(1 + v₂² /c²)

p₁ = m v₁ /√(1 + v₁² / c²)

let's look for the relationship between these two moments

p₂ / p₁ = mv₂ / mv₁ [√ (1+ v₁² / c²) /√ (1 + v₂² / c²)

from the initial statement

p₂ / p₁ = 2 √(c² + v₁²) / (c² + v₂²)

we take c from the root

p₂ / p₁ = 2 √ [(1+ v₁²) / (1 + v₂²)]

this is the exact result, to have an approximate shape suppose that the velocities are much greater than 1

p₂ / p₁ = 2 √ [v₁² / v₂²] = 2 √ [(v₁ / v₂)²]

p₂ / p₁ = 2 (v₁ / v₂)

we see the value of the moment depends on the speed of the particles