Ch 36: Special Relativity

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796당신이 사용하는 게 아니라요?교과서 변경

Knight Calc 5th Edition

Ch 36: Special Relativity

문제 54

36장, 문제 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?

검증된 단계별 안내

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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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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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