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 50a

Chapter 36, Problem 50a

In an attempt to reduce the extraordinarily long travel times for voyaging to distant stars, some people have suggested traveling at close to the speed of light. Suppose you wish to visit the red giant star Betelgeuse, which is 430 ly away, and that you want your 20,000 kg rocket to move so fast that you age only 20 years during the round trip. How fast, as a fraction of c, must the rocket travel relative to earth?

Verified step by step guidance

Step 1: Understand the problem. The goal is to determine the speed of the rocket, as a fraction of the speed of light (c), such that the astronaut ages only 20 years during the round trip to Betelgeuse, which is 430 light-years (ly) away. This involves relativistic time dilation, where the proper time experienced by the astronaut is shorter than the time observed on Earth.

Step 2: Use the time dilation formula from special relativity:

t = \frac{t}{0_{\sqrt 1 - v^{2 c^{2}}}}

, where

t

is the time observed on Earth,

t_{0}

is the proper time experienced by the astronaut,

v

is the rocket's velocity, and

c

is the speed of light.

Step 3: Calculate the total time observed on Earth for the round trip. Since Betelgeuse is 430 ly away, the round trip distance is

860

ly. The time observed on Earth is given by

\frac{860}{v}

, where

v

is the rocket's velocity.

Step 4: Relate the proper time to the observed time using the time dilation formula. Substitute

t_{0}

= 20 years and

t

\frac{860}{v}

into the time dilation formula:

\frac{860}{v} = \frac{20}{\sqrt{1 - v^{2 c^{2}}}}

Step 5: Solve for

\frac{v}{c}

. Rearrange the equation to isolate

\frac{v}{c}

, and solve algebraically. This will involve squaring both sides, simplifying, and solving for

v

as a fraction of

c

. Ensure the solution satisfies the relativistic constraints.

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Key Concepts

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

Relativity of Time

According to Einstein's theory of relativity, time is not absolute and can vary for observers in different frames of reference. When an object moves at speeds close to the speed of light, time dilation occurs, meaning that time passes more slowly for the moving object compared to a stationary observer. This concept is crucial for understanding how astronauts could age less than people on Earth during long space journeys.

Speed of Light (c)

The speed of light in a vacuum, denoted as 'c', is approximately 299,792 kilometers per second. It is the ultimate speed limit in the universe, according to relativity, meaning that no object with mass can reach or exceed this speed. Understanding this concept is essential for calculating the required speed of the rocket as a fraction of 'c' to achieve the desired aging effect during the journey.

Distance and Time Relationship in Space Travel

In space travel, the relationship between distance, speed, and time is governed by the equation distance = speed × time. For interstellar travel, this relationship must account for relativistic effects, particularly when traveling at significant fractions of the speed of light. This concept helps in determining how long the journey will take from both the perspective of the travelers and observers on Earth.

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Textbook Question

The quantity dE/dv, the rate of increase of energy with speed, is the amount of additional energy a moving object needs per 1 m/s increase in speed. A 25,000 kg rocket is traveling at 0.90c. How much additional energy is needed to increase its speed by 1 m/s?

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Textbook Question

The star Alpha goes supernova. Ten years later and 100 ly away, as measured by astronomers in the galaxy, star Beta explodes. An alien spacecraft passing through the galaxy finds that the distance between the two explosions is 120 ly. According to the aliens, what is the time between the explosions?

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Textbook Question

Two events in reference frame S occur 10 μs apart at the same point in space. The distance between the two events is 2400 m in reference frame S'. What is the velocity of S' relative to S?

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

The Stanford Linear Accelerator (SLAC) accelerates electrons to v = 0.99999997c in a 3.2-km-long tube. If they travel the length of the tube at full speed (they don’t, because they are accelerating), how long is the tube in the electrons’ reference frame?

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Textbook Question

The star Delta goes supernova. One year later and 2.0 ly away, as measured by astronomers in the galaxy, star Epsilon explodes. Let the explosion of Delta be at x_D = 0 and t_D = 0. The explosions are observed by three spaceships cruising through the galaxy in the direction from Delta to Epsilon at velocities v₁ = 0.30c, v₂ = 0.50c, and v₃ = 0.70c. All three spaceships, each at the origin of its reference frame, happen to pass Delta as it explodes. What are the times of the two explosions as measured by scientists on each of the three spaceships?

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