A proton is accelerated to 0.999c. By what factor does the proton's momentum exceed its Newtonian momentum?

Verified step by step guidance

1. Understand the problem: The question asks us to compare the relativistic momentum of a proton moving at 0.999c (where c is the speed of light) to its classical (Newtonian) momentum. This involves using the relativistic momentum formula and the classical momentum formula.
2. Write the formula for relativistic momentum: \( p_{rel} = \frac{mv}{\sqrt{1 - \frac{v^2}{c^2}}} \), where \( m \) is the mass of the proton, \( v \) is its velocity, and \( c \) is the speed of light.
3. Write the formula for Newtonian momentum: \( p_{newt} = mv \), where \( m \) is the mass of the proton and \( v \) is its velocity.
4. Set up the ratio of relativistic momentum to Newtonian momentum: \( \text{Factor} = \frac{p_{rel}}{p_{newt}} = \frac{\frac{mv}{\sqrt{1 - \frac{v^2}{c^2}}}}{mv} = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \). Notice that the mass \( m \) and velocity \( v \) cancel out.
5. Substitute \( v = 0.999c \) into the expression \( \text{Factor} = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \) to calculate the factor by which the relativistic momentum exceeds the Newtonian momentum. Simplify the denominator to find the result.

Key Concepts

Relativistic Momentum

In relativistic physics, momentum is defined as p = γmv, where γ (gamma) is the Lorentz factor, given by γ = 1 / √(1 - v²/c²). As an object's speed approaches the speed of light (c), γ increases significantly, leading to a much larger momentum than predicted by classical mechanics. This concept is crucial for understanding how particles behave at high velocities.

Newtonian Momentum

Newtonian momentum is defined as p = mv, where m is the mass and v is the velocity of an object. This formula applies well at low speeds, where relativistic effects are negligible. However, it fails to accurately describe the behavior of objects moving at speeds close to the speed of light, necessitating the use of relativistic momentum for such scenarios.

Lorentz Factor

The Lorentz factor (γ) is a crucial component in the equations of special relativity, representing the factor by which time, length, and relativistic mass increase as an object approaches the speed of light. It is calculated as γ = 1 / √(1 - v²/c²). Understanding this factor is essential for calculating relativistic effects, including the increase in momentum for particles like protons moving at relativistic speeds.