Ch. 36 - The Special Theory of Relativity

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Non è quello che usi tu?Cambia libro di testo

Giancoli Douglas 5th edition

Ch. 36 - The Special Theory of Relativity

Problema 10b

Capitolo 35, Problema 10b

A star is 23.5 light-years from Earth. How long would it take a spacecraft traveling 0.950c to reach that star as measured by observers on the spacecraft?

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Determine the distance to the star in the Earth's frame of reference, which is given as 23.5 light-years. This is the proper distance since it is measured in the rest frame of the Earth.

Identify the speed of the spacecraft as 0.950c, where c is the speed of light. This is the velocity relative to the Earth.

To calculate the time as measured by observers on the spacecraft, use the concept of time dilation from special relativity. The time dilation formula is:

t = t e_{0} / \sqrt{1 - v^{2} / c^{2}}

, where

t

is the time in the spacecraft's frame,

t_{e}

is the time in the Earth's frame, and

v

is the velocity of the spacecraft.

First, calculate the time in the Earth's frame of reference using the formula:

t_{e} = d / v

, where

d

is the distance (23.5 light-years) and

v

is the velocity (0.950c).

Substitute the value of

t_{e}

into the time dilation formula to find the time as measured by observers on the spacecraft. Simplify the expression to complete the calculation.

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Relativity of Time

According to Einstein's theory of relativity, time is not absolute and can vary for observers in different frames of reference. For a spacecraft traveling at a significant fraction of the speed of light, time experienced by those on board (proper time) will be less than that measured by stationary observers on Earth due to time dilation.

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04:27

Intro to Relative Motion (Relative Velocity)

Speed of Light and 'c'

The speed of light in a vacuum, denoted as 'c', is approximately 299,792 kilometers per second. In the context of relativity, as an object approaches the speed of light, its relativistic effects become significant, affecting time, length, and mass. The spacecraft in the question travels at 0.950c, which is 95% of the speed of light.

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06:03

The Doppler Effect (Light)

Distance and Travel Time Calculation

To calculate the time it takes to travel a certain distance at a given speed, the formula time = distance/speed is used. However, when dealing with relativistic speeds, one must also account for time dilation, which alters the perceived travel time for observers on the spacecraft compared to those on Earth.

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04:43

Calculating Displacement from Velocity-Time Graphs

Pratica correlata

Domanda del libro di testo

(I) An observer on Earth sees an alien vessel approach at a speed of 0.70c. The fictional starship Enterprise comes to the rescue (Fig. 36–17), overtaking the aliens while moving directly toward Earth at a speed of 0.90c relative to Earth. What is the relative speed of one vessel as seen by the other?

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Domanda del libro di testo

Two spaceships leave Earth in opposite directions, each with a speed of 0.50c with respect to Earth.

(a) What is the velocity of spaceship 1 relative to spaceship 2?

(b) What is the velocity of spaceship 2 relative to spaceship 1?

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Domanda del libro di testo

(I) Repeat Problem 19 using the Lorentz transformation and a relative speed v = 1.60 x 10⁸ m/s, but choose the time t to be (a) 3.5μs and (b) 10.0 μs .

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