Ch 37: Special Relativity

Young & Freedman Calc - University Physics 15th Edition

Young & Freedman Calc15th EditionUniversity PhysicsISBN: 9780135159552Not the one you use?Change textbook

Young & Freedman Calc 15th Edition

Ch 37: Special Relativity

Problem 37

Chapter 36, Problem 37

Compute the kinetic energy of a proton (mass $1.67 \times 10^{- 27}$ kg) using both the nonrelativistic and relativistic expressions, and compute the ratio of the two results (relativistic divided by nonrelativistic) for speeds of (a) $8.00 \times 10^{7}$ m/s and (b) $2.85 \times 10^{8}$ m/s.

Verified step by step guidance

Step 1: Understand the concept of kinetic energy. Kinetic energy is the energy that an object possesses due to its motion. The nonrelativistic kinetic energy (KE) is given by the formula:

K E = \frac{1}{2} m v^{2}

, where

m

is the mass and

v

is the velocity.

Step 2: Calculate the nonrelativistic kinetic energy for the proton at speed (a) 8.00 * 10^7 m/s. Use the formula:

K E = \frac{1}{2} m v^{2}

. Substitute

m = 1.67 \times 10^{- 27}

kg and

v = 8.00 \times 10^{7}

m/s.

Step 3: Calculate the relativistic kinetic energy using the formula:

K E = m c (^{2)} (\sqrt{1 - \frac{v^{2}}{c^{2}}} - 1)

, where

c

is the speed of light (

3.00 \times 10^{8}

m/s). Substitute the values for

m

and

v

as in Step 2.

Step 4: Repeat Steps 2 and 3 for speed (b) 2.85 * 10^8 m/s. Substitute

v = 2.85 \times 10^{8}

m/s into both the nonrelativistic and relativistic kinetic energy formulas.

Step 5: Compute the ratio of the relativistic kinetic energy to the nonrelativistic kinetic energy for both speeds (a) and (b). This involves dividing the relativistic result by the nonrelativistic result for each speed.

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Key Concepts

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

Kinetic Energy

Kinetic energy is the energy possessed by an object due to its motion, calculated using the formula KE = 0.5 * m * v^2 for nonrelativistic speeds. It is a measure of the work needed to accelerate an object from rest to its current velocity, and is dependent on both mass and velocity.

Relativistic Kinetic Energy

Relativistic kinetic energy accounts for the effects of special relativity at high speeds, close to the speed of light. It is calculated using the formula KE = (γ - 1) * m * c^2, where γ is the Lorentz factor, m is mass, and c is the speed of light. This formula reflects how energy increases significantly as an object's speed approaches the speed of light.

Lorentz Factor

The Lorentz factor, denoted as γ, is a crucial component in relativistic physics, defined as γ = 1 / sqrt(1 - v^2/c^2). It quantifies the amount of time dilation, length contraction, and relativistic mass increase experienced by an object moving at velocity v relative to the speed of light c. It becomes significant when v approaches c, affecting calculations of relativistic kinetic energy.

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Related Practice

Textbook Question

A particle has rest mass $6.64 \times 10^{- 27}$ kg and momentum $2.10 \times 10^{- 18}$ kgm/s.

(a) What is the total energy (kinetic plus rest energy) of the particle?

(b) What is the kinetic energy of the particle?

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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?

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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?

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Textbook Question

An observer in frame S′ is moving to the right (+x-direction) at speed u = 0.600c away from a stationary observer in frame S. The observer in S′ measures the speed v′ of a particle moving to the right away from her. What speed v does the observer in S measure for the particle if (a) v′ = 0.400c; (b) v′ = 0.900c; (c) v′ = 0.990c?

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