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Chapter 37: Problem 4

A spaceship flies past Mars with a speed of 0.985\(c\) relative to the surface of the planet. When the spaceship is directly overhead, a signal light on the Martian surface blinks on and then off. An observer on Mars measures that the signal light was on for 75.0\(\mu\) s. (a) Does the observer on Mars or the pilot on the spaceship measure the proper time? (b) What is the duration of the light pulse measured by the pilot of the spaceship?

Short Answer

Expert verified
(a) The observer on Mars measures the proper time. (b) The pilot measures the light pulse duration as approximately 434.78 渭s.

Step by step solution

01

Understanding Proper Time

Proper time is the time interval measured by an observer who sees two events occur at the same location. In this scenario, the event (light blinking on and off) is occurring at a fixed location on the Martian surface. Therefore, the observer on Mars measures the proper time.
02

Identify Given Data

The speed of the spaceship is given as 0.985c, where c is the speed of light. The proper time measured by the observer on Mars is 75.0 \( \mu s \).
03

Use Time Dilation Formula

Time dilation can be calculated using the formula: \[ t = \frac{t_0}{\sqrt{1-v^2/c^2}} \] where \( t \) is the dilated time (time measured by the moving observer), \( t_0 \) is the proper time, and \( v \) is the relative velocity (0.985c).
04

Solve for Dilated Time

Plug the values into the time dilation formula: \[ t = \frac{75.0}{\sqrt{1-(0.985)^2}} \]. Calculate the result to find the duration of the light pulse measured by the pilot.
05

Calculate the Numerical Result

First, calculate the denominator: \( \sqrt{1-(0.985)^2} = \sqrt{1-0.970225} = \sqrt{0.029775} \approx 0.1725 \). Now calculate \( t = \frac{75.0}{0.1725} \approx 434.78 \mu s \).

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Key Concepts

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

Proper Time
In the context of relativity, "proper time" is the time measured by an observer at rest relative to the event being observed. This means if two events, like a light blinking on and off, occur at the same location according to one observer, that observer measures the proper time. In the given exercise about the Martian spaceship, the observer on Mars is stationary relative to the light blinking. Therefore, he measures the proper time of the event because both the turning on and off of the signal light happen at the same place from his viewpoint.
  • Proper time is always measured by an observer at rest with respect to the event.
  • It represents the shortest time interval between two events as perceived by any observer.
  • In the problem, 75.0 \(\mu s\) is the proper time as measured by the Martian observer.
Relative Velocity
Relative velocity is a measure of how fast one object is moving in relation to another. In the realm of special relativity, relative velocity can powerfully affect time and space perceptions for observers in different frames. In the provided exercise, the relative velocity is how fast the spaceship is moving compared to Mars' surface. This speed is given as 0.985 times the speed of light, denoted as 0.985\(c\).
  • Relative velocity affects the measurement of time intervals between events.
  • For spacecraft moving at a significant fraction of the speed of light, relativistic effects like time dilation become significant.
  • Overall, the spaceship's speed relative to Mars is what causes the observer in the spaceship to experience a different time interval for the blink of the light.
Speed of Light
The speed of light, denoted by \(c\), is a critical constant in physics, approximately equal to 299,792,458 meters per second. Not only is it the fastest speed at which information or matter can travel, but it also forms the backbone of Einstein's theory of relativity. In our exercise, both the spaceship's speed as 0.985\(c\) and the resulting time dilation computed are based upon the invariance of the speed of light.
  • The speed of light remains constant across all frames of reference.
  • This invariant speed factor is what leads to phenomena such as time dilation.
  • In the exercise, the calculations depend heavily on using \(c\) to determine how time appears to slow down for the fast-moving spaceship observer.
Special Relativity
Special relativity is a theory formulated by Einstein that revolutionized how we understand time, space, and motion. It tells us that observers in different inertial frames may perceive time and space differently, especially when moving close to the speed of light. This exercise specifically applies principles from special relativity, like time dilation, which alter how different observers measure the interval of a single event.
  • Special relativity explains how measures of time and space vary based on the observer's velocity.
  • Time dilation is a key concept within this theory, pivotal for space travel scenarios.
  • In the Martian spaceship scenario, this effect of special relativity leads to different times being recorded by Mars' observer and the spaceship pilot.

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Most popular questions from this chapter

One of the wavelengths of light emitted by hydrogen atoms under normal laboratory conditions is \(\lambda=656.3 \mathrm{nm},\) in the red portion of the electromagnetic spectrum. In the light emitted from a distant galaxy this same spectral line is observed to be Doppler- shifted to \(\lambda=953.4 \mathrm{nm},\) in the infrared portion of the spectrum. How fast are the emitting atoms moving relative to the earth? Are they approaching the earth or receding from it?

As you pilot your space utility vehicle at a constant speed toward the moon, a race pilot fies past you in her spaceracer at a constant speed of 0.800\(c\) relative to you. At the instant the spaceracer passes you, both of you start timers at zero. (a) At the instant when you measure that the spaceracer has traveled \(1.20 \times 10^{8} \mathrm{m}\) past you, what does the race pilot read on her timer? (b) When the race pilot reads the value calculated in part (a) on her timer, what does she measure to be your distance from her? (c) At the instant when the race pilot reads the value calculated in part (a) on her timer, what do you read on yours?

The starships of the Solar Federation are marked with the symbol of the federation, a circle, while starships of the Denebian Empire are marked with the empire's symbol, an ellipse whose major axis is 1.40 times longer than its minor axis \((a=1.40 b\) in Fig. P37.51). How fast, relative to an observer, does an empire ship have to travel for its marking to be confused with the marking of a federation ship?

The positive muon \(\left(\mu^{+}\right),\) an unstable particle, lives on average \(2.20 \times 10^{-6} \mathrm{s}\) (measured in its own frame of reference) before decaying. (a) If such a particle is moving, with respect to the laboratory, with a speed of \(0.900 c,\) what average lifetime is measured in the laboratory? (b) What average distance, measured in the laboratory, does the particle move before decaying?

A cube of metal with sides of length \(a\) sits at rest in a frame \(S\) with one edge parallel to the \(x\) -axis. Therefore, in \(S\) the cube has volume \(a^{3} .\) Frame \(S^{\prime}\) moves along the \(x\) -axis with a speed \(u\) . As measured by an observer in frame \(S^{\prime},\) what is the volume of the metal cube?
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