Ch 36: Special Relativity

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 36: Special Relativity

Problem 54

Chapter 36, Problem 54

Two rockets approach each other. Each is traveling at 0.75c in the earth's reference frame. What is the speed, as a fraction of c, of one rocket relative to the other?

Verified step by step guidance

Step 1: Recognize that this problem involves relative velocity in the context of special relativity. The classical formula for relative velocity does not apply when speeds are close to the speed of light. Instead, use the relativistic velocity addition formula:

v_{rel} = \(\frac{v_1 + v_2}{1 + \frac{v_1 v_2}{c^2}\)}

, where

v_1

and

v_2

are the velocities of the two objects in the same reference frame, and

c

is the speed of light.

Step 2: Assign the given values to the formula. In the earth's reference frame, both rockets are traveling at

0.75c

, but in opposite directions. To account for this, one velocity will be positive (

v_1 = 0.75c

) and the other negative (

v_2 = -0.75c

Step 3: Substitute

v_1 = 0.75c

and

v_2 = -0.75c

into the relativistic velocity addition formula:

v_{rel} = \(\frac{0.75c - 0.75c}{1 - \frac{(0.75c)(-0.75c)}{c^2}\)}

Step 4: Simplify the numerator and denominator of the formula. The numerator becomes

1.5c

, and the denominator involves the product of

0.75 \(\times\) 0.75

, divided by

c^2

. Carefully simplify the denominator to ensure accuracy.

Step 5: After simplifying, the result will yield the relative velocity

v_{rel}

as a fraction of

c

. This fraction represents the speed of one rocket relative to the other in the earth's reference frame.

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Key Concepts

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

Relativistic Velocity Addition

In special relativity, the formula for adding velocities differs from classical mechanics. When two objects move at relativistic speeds (close to the speed of light), their relative velocity is calculated using the formula: v' = (u + v) / (1 + uv/c²), where u and v are the velocities of the two objects, and c is the speed of light. This ensures that the resultant speed never exceeds the speed of light.

Reference Frames

A reference frame is a perspective from which motion is observed and measured. In this problem, the Earth serves as the reference frame for the rockets' speeds. Understanding how different reference frames affect the perception of speed is crucial in relativity, as speeds can appear different depending on the observer's frame of reference.

Speed of Light (c)

The speed of light in a vacuum, denoted as c, is a fundamental constant in physics, approximately 3.00 x 10^8 meters per second. It serves as the ultimate speed limit in the universe, meaning no object with mass can reach or exceed this speed. In relativistic physics, speeds are often expressed as fractions of c to simplify calculations and comparisons.

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Related Practice

Textbook Question

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The half-life of a muon at rest is 1.5 μs. Muons that have been accelerated to a very high speed and are then held in a circular storage ring have a half-life of 7.5 μs. What is the total energy of a muon in the storage ring? The mass of a muon is 207 times the mass of an electron.

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Textbook Question

Derive a velocity transformation equation for u_y and u'_y. Assume that the reference frames are in the standard orientation with motion parallel to the x- and x'-axes.

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Textbook Question

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Textbook Question

A rocket traveling at 0.50c sets out for the nearest star, Alpha Centauri, which is 4.3 ly away from earth. It will return to earth immediately after reaching Alpha Centauri. What distance will the rocket travel and how long will the journey last according to (a) stay-at-home earthlings and (b) the rocket crew? (c) Which answers are the correct ones, those in part a or those in part b?

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Textbook Question

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