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Chapter 14: Problem 7

The premise of the Planet of the Apes movies and book is that hibernating astronauts travel far into Earth's future, to a time when human civilization has been replaced by an ape civilization. Considering only special relativity, determine how far into Earth's future the astronauts would travel if they slept for \(120 \mathrm{y}\) whilc traveling relative to Earth with a speed of \(0.9990 \mathrm{c},\) first outward from Earth and then back again.

Short Answer

Expert verified
The astronauts travel approximately 2683 years into Earth's future.

Step by step solution

01

Understand the Problem

We are asked to determine how far into the future astronauts would travel if they move at a speed of 0.9990c (where c is the speed of light) away from and back to Earth, effectively sleeping for 120 years during the journey. This involves using the concepts of time dilation from Einstein's theory of special relativity.
02

Recall the Time Dilation Formula

In special relativity, time dilation can be calculated using the equation: \( t' = \frac{t}{\sqrt{1 - \left( \frac{v}{c} \right)^2 }} \), where \( t' \) is the time experienced by stationary observers (on Earth), \( t \) is the proper time experienced by the traveling astronauts, \( v \) is their velocity, and \( c \) is the speed of light.
03

Set Up the Equation

In this scenario, the astronauts experience 120 years. Substitute \( t = 120 \mathrm{y} \) and \( v = 0.9990c \) into the time dilation formula to solve for \( t' \).
04

Calculate the Time Dilation Factor

First, calculate the dilation factor \( \frac{1}{\sqrt{1 - \left( \frac{0.9990c}{c} \right)^2 }} \). This involves calculating the square of the velocity fraction: \( \left( \frac{0.9990c}{c} \right)^2 = 0.9990^2 = 0.998001 \). Then, calculate the square root of \( 1 - 0.998001 \) to find the dilation factor.
05

Solve for Earth's Time \( t' \)

Compute \( t' \) using the dilation factor calculated in the previous step: \( t' = \frac{120}{\sqrt{1 - 0.998001}} \). Simplify \( \sqrt{1 - 0.998001} \approx 0.0447 \) and find the result for \( t' \) which gives Earth's observer's time.
06

Calculate Total Return Time

Since the journey involves traveling away and then back to Earth, calculate the total time by doubling the result from Step 5. This gives the total Earth time experienced by the astronauts over their entire journey.

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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 key concept in Einstein's theory of special relativity. It refers to the difference in the elapsed time measured by two observers, due to a relative velocity between them. In simpler terms, if you travel at a very high speed, close to the speed of light, time will seem to move slower for you compared to someone who remains stationary.
Imagine you onboard a spaceship traveling at 99.90% of the speed of light, as in our scenario. For you, 120 years might pass, but for an observer on Earth, a much longer time will elapse due to time dilation. This effect becomes significant only at speeds approaching the speed of light.
  • High speeds lead to a noticeable time dilation.
  • Proper time is what the traveling observer measures.
  • Dilated time is what a stationary observer measures.
This concept is crucial for futuristic scenarios like long-distance space travel and has exciting implications, as we'll explore in related concepts.
Einstein's Theory
Albert Einstein's theory of special relativity, formulated in 1905, revolutionized our understanding of space and time. It fundamentally changed how we perceive motion for objects traveling at or near the speed of light.
This theory introduces two main postulates:
  • The laws of physics are the same in all inertial frames of reference.
  • The speed of light is constant in a vacuum for all observers, regardless of their motion relative to the light source.
From these postulates, several surprising results emerge, including time dilation and length contraction. The notion that the faster you move, the slower time moves for you relative to stationary observers, is central to the idea of time dilation, as previously noted.
Special relativity provides a framework for understanding how time and space are linked, bringing us insights into high-speed travel and its fascinating consequences.
Speed of Light
The speed of light, denoted by the symbol \(c\), is a fundamental constant in physics. It measures approximately \(299,792,458\) meters per second or about \(186,282\) miles per second. This speed is not just a speed limit for light but for all massless particles and influences how we understand the universe's structure.
Since nothing with mass can exceed this speed, it plays a central role in Einstein's theories. When objects approach the speed of light, as in our astronaut scenario moving at 0.9990c, effects like time dilation and mass increase come into play.
  • The universal speed limit inherent in nature.
  • Dictates the experiences of time and space under relativistic speeds.
  • This constant is essential for all fundamental laws of electromagnetism and relativity.
Understanding the speed of light helps us grasp the fundamental limits of travel and communication, crucial for contemplating high-speed interstellar travel.
Future Time Travel
The intriguing idea of traveling into the future, as depicted in our initial scenario, is a practical outcome of special relativity. While it doesn't involve futuristic machines, it relies on the relativistic effects experienced during high-speed travel.
Time travelers, like our astronauts, effectively leap forward in time as perceived by stationary observers. This is due to the time dilation that occurs at such extreme speeds. While the astronauts experience only 120 years of time aboard their spacecraft, many more years pass on Earth.
  • Based on high-speed relativity rather than mythical time machines.
  • Relies on the principles and effects of special relativity.
  • An exciting concept with real-world implications in the future of space travel.
In this way, special relativity offers a scientifically grounded means to explore the concept of future time travel, with its potential realized through technology that could one day bring interstellar voyages to reality.

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Most popular questions from this chapter

The center of our Milky Way galaxy is about 23000 ly away. (a) To cight significant figures, at what constant speed parameter would you need to travel exactly 23000 ly (measured in the Galaxy frame) in cxactly \(30 \mathrm{y}\) (measured in your frame)? (b) Measured in your frame and in lightyears, what length of the Galaxy would pass by you during the trip?

A venturi meter is used to measure the flow speed of a fluid in a pipe. The meter is connected between two sections of the pipc (Fig. \(14-50) ;\) the cross-scetional arca \(A\) of the entrance and exit of the meter matches the pipe's cross-sectional arca. Between the cntrance and exit, the fluid flows from the pipe with speed \(V\) and then through a narrow "throat" of cross- scetional area \(a\) with speed \(v\). A manometer connects the wider portion of the meter to the narrower portion. The change in the fluid's specd is accompanied by a change \(\Delta p\) in the fluid's pressure, which causes a height difference \(h\) of the liquid in the two arms of the manomcter. (Here \(\Delta p\) means pressure in the throat minus pressurc in the pipe.) (a) By applying Bernoulli's equation and the equation of continuity to points 1 and 2 in Fig. \(14-50,\) show that $$ V=\sqrt{\frac{2 a^{2} \Delta p}{\rho\left(a^{2}-A^{2}\right)}} $$ where \(\rho\) is the density of the fluid. (b) Suppose that the fluid is fresh water, that the cross-scctional arcas are \(64 \mathrm{~cm}^{2}\) in the pipe and \(32 \mathrm{~cm}^{2}\) in the throat, and that the pressure is \(55 \mathrm{kPa}\) in the pipe and \(41 \mathrm{kPa}\) in the throat. What is the rate of water flow in cubic meters per second?

Space cruisers \(A\) and \(B\) are moving parallel to the positive direction of an \(x\) axis. Cruiser \(A\) is faster, with a relative speed of \(v=0.900 c,\) and has a proper length of \(L=200 \mathrm{~m}\). According to the pilot of \(A\), at the instant \((t=0)\) the tails of the cruisers are aligned, the noses are also. According to the pilot of \(B,\) how much later are the noses aligned?

Inertial frame \(S^{\prime}\) moves at a speed of \(0.60 \mathrm{c}\) with respect to frame \(S\) (Fig. 37-9). Further, \(x=x^ {\prime}=0\) at \(t=t^{\prime}=0 .\) Two cvents are recorded. In frame \(S\), event 1 occurs at the origin at \(t=0\) and event 2 occurs on the \(x\) axis at \(x=3,0 \mathrm{~km}\) at \(t=4.0 \mathrm{prs}\). According to observer \(S^{\prime},\) what is the time of (a) event 1 and (b) event 27 (c) Do the two observers see the same sequence or the reverse?

What is the speed parameter for the following speeds: (a) a typical rate of continental drift ( 1 in \(/ y\) ); (b) a typical drift speed for electrons in a current-carrying conductor \((0.5 \mathrm{~mm} / \mathrm{s}) ;\) (c) a highway speed limit of \(55 \mathrm{mi} / \mathrm{h} ;\) (d) the root-mean-square speed of a hydrogen molccule at room temperature; (c) a supersonic plane flying at Mach \(2.5(1200 \mathrm{~km} / \mathrm{h}) ;\) (f) the escape speed of a projectile from the Earth's surface; (g) the speed of Earth in its orbit around the Sun; (h) a typical recession speed of a distant quasar due to the cosmological expansion \(\left(3.0 \times 10^{4} \mathrm{~km} / \mathrm{s}\right) ?\)
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