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 14th Edition

Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook

All textbooks

Young & Freedman Calc 14th Edition

Ch 37: Special Relativity

Problem 27

Chapter 37, 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?

Verified step by step guidance

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 $.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

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.

Recommended video:

Guided course

05:33

Calculating Change in Velocity from Acceleration-Time Graphs

Related Practice

Textbook Question

A proton (rest mass $1.67 \times 10^{- 27}$ kg) has total energy that is $4.00$ times its rest energy. What are (a) the kinetic energy of the proton; (b) the magnitude of the momentum of the proton; (c) the speed of the proton?

views

Textbook Question

Electrons are accelerated through a potential difference of $750$ kV, so that their kinetic energy is $7.50 \times 10^{5}$ eV.

(a) What is the ratio of the speed $v$ of an electron having this energy to the speed of light, $c$ ?

(b) What would the speed be if it were computed from the principles of classical mechanics?

views

Textbook Question

Tell It to the Judge. (a) How fast must you be approaching a red traffic light (λ = 675 nm) for it to appear yellow (λ = 575 nm)? Express your answer in terms of the speed of light. (b) If you used this as a reason not to get a ticket for running a red light, how much of a fine would you get for speeding? Assume that the fine is \$1.00 for each kilometer per hour that your speed exceeds the posted limit of 90 km/h.

views

Textbook Question

A force is applied to a particle along its direction of motion. At what speed is the magnitude of force required to produce a given acceleration twice as great as the force required to produce the same acceleration when the particle is at rest? Express your answer in terms of the speed of light.

views

Textbook Question

Relativistic Baseball. Calculate the magnitude of the force required to give a 0.145 kg baseball an acceleration a = 1.00 m/s² in the direction of the baseball's initial velocity when this velocity has a magnitude of (a) 10.0 m/s; (c) 0.990c.

views

Textbook Question

A source of electromagnetic radiation is moving in a radial direction relative to you. The frequency you measure is 1.25 times the frequency measured in the rest frame of the source. What is the speed of the source relative to you? Is the source moving toward you or away from you?

views