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Chapter 37: Problem 14

A rocket ship flies past the earth at 85.0\(\%\) of the speed of light. Inside, an astronaut who is undergoing a physical examination is having his height measured while he is lying down parallel to the direction the rocket ship is moving. (a) If his height is measured to be 2.00 \(\mathrm{m}\) by his doctor inside the ship, what height would a person watching this from earth measure for his height? (b) If the earth-based person had measured \(2.00 \mathrm{m},\) what would the doctor in the spaceship have measured for the astronaut's height? Is this a reasonable height? (c) Suppose the astronaut in part (a) gets upafter the examination and stands with his body perpendicular to the direction of motion. What would the doctor in the rocket and the observer on earth measure for his height now?

Short Answer

Expert verified
(a) 1.053 m; (b) 3.80 m, which is unrealistic; (c) 2.00 m.

Step by step solution

01

Introduction of Concepts

For this exercise, we apply the concept of length contraction, which is a relativistic effect that occurs at high speeds. According to the theory of relativity, an object in motion contracts in the direction of motion relative to an observer at rest. The formula for length contraction is: \[ L' = L \sqrt{1 - \frac{v^2}{c^2}} \] where \(L'\) is the contracted length, \(L\) is the proper length (length in the rest frame), \(v\) is the velocity of the object, and \(c\) is the speed of light.
02

Determine Length Contraction in Part (a)

Given: - Proper length \( L = 2.00 \, \text{m} \) (measured in the astronaut's frame)- Speed \( v = 0.85c \)Substitute these into the length contraction formula:\[ L' = 2.00 \, \text{m} \times \sqrt{1 - (0.85)^2} \] Calculating inside the square root: \[ 1 - (0.85)^2 = 1 - 0.7225 = 0.2775 \] \[ \sqrt{0.2775} \approx 0.5265 \]Now, calculate \( L' \):\[ L' = 2.00 \, \text{m} \times 0.5265 \approx 1.053 \, \text{m} \]Thus, an earth-based observer would measure the astronaut's height as approximately 1.053 meters.
03

Reverse the Perspective in Part (b)

If the earth-based person measures the astronaut's height as 2.00 meters, this means 2.00 meters is the contracted length. We need to find the proper length as measured by the spaceship doctor.Rearrange the length contraction formula: \[ L = \frac{L'}{\sqrt{1 - \frac{v^2}{c^2}}} \]Substituting \( L' = 2.00 \, \text{m} \) and \( v = 0.85c \):\[ L = \frac{2.00 \, \text{m}}{\sqrt{0.2775}} \approx \frac{2.00 \, \text{m}}{0.5265} \approx 3.80 \, \text{m} \]So, the doctor would measure the astronaut's height as 3.80 meters.
04

Assess Reasonableness in Part (b)

The height of 3.80 meters is unusually large for a human, indicating the impact of relativistic effects when measurements have been improperly switched between frames.
05

Measure Perpendicular Height in Part (c)

When the astronaut stands up, his height is measured perpendicular to the direction of motion, so no relativistic length contraction occurs. Whether by the doctor on the spaceship or by an earth observer, the astronaut's height remains 2.00 meters.

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Key Concepts

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

Relativity
Relativity, developed by Albert Einstein, revolutionized our understanding of space, time, and how they interact. One key idea is that the laws of physics are the same for all non-accelerating observers, regardless of their state of motion. This principle is known as the principle of relativity. A fascinating outcome of relativity is that observations of space and time can change depending on the observer's relative motion. The famous equation, \(E=mc^2\), represents the relationship between mass and energy, highlighting that mass can be transformed into energy and vice versa.
Relativity also tells us that the speed of light is always constant within any frame of reference, no matter how fast the observer moves. This constancy leads to bizarre effects such as time dilation and length contraction, which are foundational concepts for understanding phenomena at high velocities close to the speed of light.
Speed of Light
The speed of light, denoted by \(c\), is a fundamental constant in physics. It has a value of approximately 299,792,458 meters per second in a vacuum. The constancy of the speed of light is a cornerstone of Einstein's theory of relativity.
This means that no matter how fast an observer is moving, they will always measure the speed of light to be the same. It's an upper limit; nothing with mass can travel faster than the speed of light. As an object approaches this speed, its relativistic mass increases, requiring infinite energy to reach \(c\), making it impossible for material objects to exceed this speed.
Light's speed affects how we interpret distances and times in the universe. Observers moving relative to each other do not agree on measurements of space and time, which leads to fascinating effects like time dilation and length contraction.
Lorentz Contraction
Lorentz Contraction, or length contraction, is a phenomenon predicted by the theory of relativity. It describes how objects appear to contract along the direction of motion when they move close to the speed of light. Mathematically, this is represented by the formula: \[ L' = L \sqrt{1 - \frac{v^2}{c^2}} \] where:
  • \(L'\) is the observed contracted length.
  • \(L\) is the proper length, the length in the object's rest frame.
  • \(v\) is the object's velocity.
  • \(c\) is the speed of light.
As \(v\) approaches \(c\), the term under the square root becomes smaller, causing \(L'\) to decrease. This means an object traveling at relativistic speeds will appear shorter to an outside observer aligned with the direction of motion. Importantly, this contraction only occurs in the direction of motion and not perpendicular to it. This concept helps explain how measurements can differ vastly depending on the relative motion of observers.
Reference Frames
A reference frame is a perspective from which an observer measures and interprets events. In relativity, reference frames can be inertial or non-inertial. An inertial frame is one that is either at rest or moves at a constant velocity. Forces do not affect it, so the familiar laws of physics, particularly Newton's first law of motion, apply without any pseudo-forces.
Non-inertial frames, by contrast, are accelerating. Observers in such frames may experience fictitious forces, like the sensation of being pushed back into a car seat when it accelerates.
In relativistic physics, different observers in different reference frames might disagree on measurements like time intervals and lengths due to the effects of relativity. Importantly, all inertial observers will agree on the laws of physics and the speed of light, even if they disagree on time and space measurements. This relativity of simultaneity means that events that occur at the same time for one observer may not for another.

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Most popular questions from this chapter

When Should You Use Relativity? As you have seen, relativistic calculations usually involve the quantity \(\gamma .\) When \(\gamma\) is appreciably greater than \(1,\) we must use relativistic formulas instead of Newtonian ones. For what speed \(v\) (in terms of \(c )\) is the value of \(\gamma(\) a) 1.0\(\%\) greater than \(1 ;(\) b) 10\(\%\) greater than \(1 ;(\mathrm{c}) 100 \%\) greater than 1\(?\)

A proton (rest mass \(1.67 \times 10^{-27} \mathrm{kg} )\) has total energy that is 4.00 times its rest energy. What are (a) the kinetic energy of the proton; (b) the magnitude of the momentum of the proton; (c) the speed of the proton?

Determining the Masses of Stars. Many of the stars in the sky are actually binary stars, in which two stars orbit about their common center of mass. If the orbital speeds of the stars are high enough, the motion of the stars can be detected by the Doppler shifts of the light they empler. Stars for which this is the case are called spectroscopic binary stars. Figure \(\mathrm{P3} 7.75\) shows the simplest case of a spectroscopic binary star: two identical stars, each with mass \(m,\) orbiting their center of mass in a circle of radius \(R\) . The plane of the stars' orbits is edge-on to the line of sight of an observer on the earth. (a) The light produced by heated hydrogen gas in a laboratory on the earth has a frequency of \(4.568110 \times\) \(10^{14}\) Hz. In the light received from the stars by a telescope on the earth, hydrogen light is observed to vary in frequency between \(4.567710 \times 10^{14} \mathrm{Hz}\) and \(4.568910 \times 10^{14} \mathrm{Hz}\) . Determine whether the binary star system as a whole is moving toward or away from the earth, the speed of this motion, and the orbital speeds of the stars. Hint: The speeds involved are much less than \(c,\) so you may use the approximate result \(\Delta f / f=u / c\) given in Section 37.6 . (b) The light from each star in the binary system varies from its maximum frequency to its minimum frequency and back again in 11.0 days. Determine the orbital radius \(R\) and the mass \(m\) of each star. Give your answer for \(m\) in kilograms and as a multiple of the mass of the sun, \(1.99 \times 10^{30}\) kg. Compare the value of \(R\) to the distance from the earth to the sun, \(1.50 \times 10^{11}\) m. (This technique is actually used in astronomy to determine the masses of stars. In practice, the problem is more complicated because the two stars in a binary system are usually not identical, the orbits are usually not circular, and the plane of the orbits is usually tilted with respect to the line of sight from the earth.)

Space Travel? Travel to the stars requires hundreds or thousands of years, even at the speed of light. Some people have suggested that we can get around this difficulty by accelerating the rocket (and its astronauts) to very high speeds so that they will age less due to time dilation. The fly in this ointment is that it takes a great deal of energy to do this. Suppose you want to go to the immense red giant Betelgeuse, which is about 500 light-years away. (A light- year is the distance that light travels in a year.) You plan to travel at constant speed in a 1000 -kg rocket ship (a little over a ton), which, in reality, is far too small for this purpose. In each case that follows, calculate the time for the trip, as measured by people on earth and by astronauts in the rocket ship, the energy needed in ioules, and the energy needed as a percentage of U.S. yearly use (which is 1.0 \(\times 10^{20} \mathrm{J} ) .\) For comparison, arrange your results in a table showing \(v_{\text { rocket }}, t_{\text { earth }}, t_{\text { rocket }}, E(\) in \(\mathrm{J}),\) and \(E\) (as \(\%\) of U.S. use). The rocket ship's speed is (a) \(0.50 \mathrm{c} ;\) (b) 0.99 \(\mathrm{c}\) ; (c) 0.9999 \(\mathrm{c} .\) On the basis of your results, does it seem likely that any government will invest in such high-speed space travel any time soon?

The French physicist Armand Fizeau was the first to measure the speed of light accurately. He also found experimentally that the speed, relative to the lab frame, of light traveling in a tank of water that is itself moving at a speed \(V\) relative to the lab frame is $$v=\frac{c}{n}+k V$$ where \(n=1.333\) is the index of refraction of water. Fizeau called \(k\) the draging coefficient and obtained an experimental value of \(k=0.44 .\) What value of \(k\) do you calculate from relativistic transformations?
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