How can we calculate (relativisticly correct) ratio ?e/?p if proton and electron have same speeds

To calculate the relativistic ratio of energy to momentum, specifically for a proton and an electron moving at the same speed, we need to delve into some fundamental concepts of relativistic physics. The energy and momentum of a particle are influenced by its mass and velocity, especially when those velocities approach the speed of light. Let's break this down step by step.

Understanding Energy and Momentum in Relativity

In the realm of special relativity, the total energy (E) and momentum (p) of a particle are given by the following equations:

• Total Energy: E = γmc²
• Momentum: p = γmv

Here, γ (gamma) is the Lorentz factor, defined as:

γ = 1 / √(1 - v²/c²)

where v is the velocity of the particle and c is the speed of light. The mass m is the rest mass of the particle, which differs for protons and electrons.

Calculating the Ratio E/p

Now, to find the ratio of energy to momentum (E/p), we can substitute the expressions for E and p:

Ratio (E/p) = (γmc²) / (γmv)

Notice that the γ factors cancel out:

Ratio (E/p) = c² / v

Applying to Proton and Electron

Since we are considering a proton and an electron moving at the same speed, we can use the same value for v in our calculations. However, we must remember that the rest masses of the proton and electron are different:

• Rest mass of proton (mp) ≈ 938 MeV/c²
• Rest mass of electron (me) ≈ 0.511 MeV/c²

Even though the ratio E/p simplifies to c²/v, the actual energy and momentum values will differ due to their respective masses. The key point is that the ratio itself, when expressed in terms of speed, remains the same for both particles at the same velocity.

Example Calculation

Let’s say both particles are moving at a speed of 0.8c. We can calculate the Lorentz factor:

γ = 1 / √(1 - (0.8c)²/c²) = 1 / √(1 - 0.64) = 1 / √0.36 = 5/3 ≈ 1.67

Now, substituting into the ratio:

Ratio (E/p) = c² / (0.8c) = c / 0.8 = 1.25c

This means that for both the proton and the electron, at the same speed, the ratio of energy to momentum is the same, approximately 1.25 times the speed of light.

Final Thoughts

In summary, while the individual energies and momenta of the proton and electron differ due to their masses, the relativistic ratio of energy to momentum remains consistent when they travel at the same speed. This illustrates a fascinating aspect of relativistic physics, where the underlying principles apply uniformly across different particles, despite their distinct properties.