A proton has momentum with magnitude p0 when its speed is 0.400c. In terms of p0, what is the magnitude of the proton's momentum when its speed is doubled to 0.800c?

Ch 37: Special Relativity

Young & Freedman Calc - University Physics 15th Edition

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Young & Freedman Calc 15th Edition

Ch 37: Special Relativity

Problem 27

Chapter 36, Problem 27

A proton has momentum with magnitude p₀ when its speed is 0.400c. In terms of p₀, what is the magnitude of the proton's momentum when its speed is doubled to 0.800c?

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Step 1: Recall the relativistic momentum formula: $ p = \frac{mv}{\sqrt{1 - \frac{v^2}{c^2}}} $, where $ p $ is the relativistic momentum, $ m $ is the rest mass, $ v $ is the velocity, and $ c $ is the speed of light.

Step 2: For the initial momentum $ p_0 $, substitute $ v = 0.400c $ into the formula: $ p_0 = \frac{m(0.400c)}{\sqrt{1 - \frac{(0.400c)^2}{c^2}}} $. Simplify the denominator to $ \sqrt{1 - 0.160} $.

Step 3: For the new momentum $ p $ when the speed is doubled to $ v = 0.800c $, substitute $ v = 0.800c $ into the formula: $ p = \frac{m(0.800c)}{\sqrt{1 - \frac{(0.800c)^2}{c^2}}} $. Simplify the denominator to $ \sqrt{1 - 0.640} $.

Step 4: Express $ p $ in terms of $ p_0 $. Notice that $ p_0 $ and $ p $ share the same rest mass $ m $, so compare the ratios of their velocities and denominators: $ p = p_0 \cdot \frac{0.800}{0.400} \cdot \frac{\sqrt{1 - 0.160}}{\sqrt{1 - 0.640}} $.

Step 5: Simplify the expression to find the relationship between $ p $ and $ p_0 $. This will give the magnitude of the proton's momentum when its speed is doubled in terms of $ p_0 $.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Momentum in Relativity

In the context of special relativity, momentum is defined as p = γmv, where γ (gamma) is the Lorentz factor, m is the rest mass, and v is the velocity. As an object's speed approaches the speed of light, its momentum increases significantly due to the relativistic effects, which must be considered when calculating momentum at high velocities.

Lorentz Factor

The Lorentz factor, γ, is given by the equation γ = 1 / √(1 - v²/c²), where v is the object's velocity and c is the speed of light. This factor accounts for time dilation and length contraction experienced by objects moving at relativistic speeds, and it plays a crucial role in determining the momentum of particles like protons as their speed increases.

Velocity Doubling and Momentum Change

When the speed of an object is doubled in a relativistic context, the change in momentum is not linear due to the Lorentz factor. For example, if a proton's speed increases from 0.400c to 0.800c, the corresponding increase in momentum must be calculated using the relativistic momentum formula, which will show that the momentum does not simply double but increases more significantly due to the effects of relativity.

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