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Chapter 26: Problem 18

The distance to Planet X from Earth is 1.00 light-year. (a) How long does it take a spaceship to reach \(X\), according to the pilot of the spaceship, if the speed of the ship is \(0.700 c\) relative to \(X ?\) (b) How long does it take the ship to make the trip according to an astronaut already stationed on Planet \(X ?\) (c) Determine the distance between Earth and Planet \(X\) according to the pilot and according to the X-based astronaut and explain why the two answers are different.

Short Answer

Expert verified
(a) 1.02 years; (b) 1.43 years; (c) 0.714 light-years for pilot, 1.00 light-year for astronaut. Pilot experiences length contraction.

Step by step solution

01

Identify Given Information

We have a distance to Planet X from Earth of 1.00 light-year. The ship's speed relative to X is given as 0.700c, where \(c\) is the speed of light. We need to find the time taken for the journey from two perspectives: the pilot's and the astronaut's on Planet X.
02

Calculate Time According to the Pilot

The time dilated due to the speed of the spaceship can be calculated using the formula of time dilation: \( t' = \frac{t}{\sqrt{1 - \frac{v^2}{c^2}}} \). But first, calculate the time as observed in the rest frame \( t = \frac{d}{v} \), where \(d = 1.00\) light-year and \(v = 0.700c\). Thus, \( t = \frac{1.00}{0.700} \approx 1.429 \) years. This is the observed time on Earth or Planet X.
03

Apply Time Dilation for Pilot's Time

Using the time dilation formula, \( t' = t \cdot \sqrt{1 - \frac{0.700^2}{1^2}} \). Calculate \[ \sqrt{1 - 0.700^2} = \sqrt{1 - 0.49} = \sqrt{0.51} \approx 0.714 \]. Therefore, \( t' = 1.429 \times 0.714 \approx 1.02 \) years. This is how long the journey takes according to the pilot.
04

Calculate Time According to the Astronaut

For the astronaut on Planet X, no relativistic effects are observed, so the time remains as calculated in Step 2, which is \( 1.429 \) years. This is the time it takes according to the astronaut on Planet X.
05

Calculate the Distance as Measured by the Pilot

Due to length contraction, the pilot measures a shorter distance. The contracted length is calculated as \( L = L_0 \times \sqrt{1 - \frac{v^2}{c^2}} \), where \(L_0 = 1.00\) light-year. Thus, \( L = 1.00 \times 0.714 \approx 0.714 \) light-years.
06

Compare the Distances and Explain Discrepance

The distance according to the astronaut on Planet X is the original 1.00 light-year. The discrepancy arises because, from the pilot's frame of reference, the distance contracts as the spaceship approaches relativistic speeds, which do not affect the astronaut's measurement.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Time Dilation
Time dilation is a fascinating concept from Einstein's theory of relativity. It describes how time is perceived differently by observers depending on their relative motion. In our exercise, this concept explains why the pilot of the spaceship and the astronaut on Planet X experience time differently during the journey.

From the perspective of the astronaut on Planet X, time flows normally, and the trip to Planet X appears to take 1.429 years, calculated simply by dividing the distance (1.00 light-year) by the speed of the spaceship (0.700c). However, for the pilot moving at 0.700 times the speed of light, time dilation occurs.
  • The pilot experiences less time due to their high speed relative to Planet X.
  • We used the time dilation formula: \( t' = t \cdot \sqrt{1 - \frac{v^2}{c^2}} \) to find out how much time the pilot would observe passing: approximately 1.02 years.
This demonstrates that moving at significant fractions of the speed of light can cause observers to experience time very differently.
Length Contraction
Length contraction is another key aspect of special relativity. It explains how the length of an object or the distance to a destination appears shorter when moving at a high velocity.

In our problem, the distance from Earth to Planet X is originally 1.00 light-year as observed by a stationary astronaut. But for the pilot moving in the spaceship, this distance is contracted.
  • The formula for length contraction is: \( L = L_0 \cdot \sqrt{1 - \frac{v^2}{c^2}} \)
  • For the spaceship traveling at 0.700c, the pilot measures the distance to be around 0.714 light-years.
This phenomenon occurs because as speed increases to relativistic levels, the dimensions along the direction of motion contract.
Hence, the pilot perceives a shorter journey compared to what the stationary observer sees.
Speed of Light
The speed of light, denoted by the symbol \( c \), plays a pivotal role in the theory of relativity. It is a constant at approximately 299,792 kilometers per second (or about 186,282 miles per second), and acts as the ultimate speed limit in the universe.

The spaceship from our exercise travels at 0.700 times the speed of light, or 0.700c. This speed choice highlights the relativistic effects like time dilation and length contraction because:
  • As the velocity of an object approaches \( c \), the effects of relativity become more pronounced.
  • Relativistic speeds significantly alter how time and space are perceived by different observers.
The concept that nothing can travel faster than the speed of light has profound implications for physics.
For instance, it ensures that causality is maintained and plays a crucial role in how we understand the structure of space-time.

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Most popular questions from this chapter

A proton moving with a constant speed has a total energy 3.5 times its rest energy. What are the proton's (a) speed, (b) kinetic energy, and (c) momentum?

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