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Chapter 37: Problem 8

An alien spacecraft is flying overhead at a great distance as you stand in your backyard. You see its searchlight blink on for 0.150 s. The first officer on the spacecraft measures that the searchlight is on for 12.0 ms. (a) Which of these two measured times is the proper time? (b) What is the speed of the spacecraft relative to the earth, expressed as a fraction of the speed of light \(c\)?

Short Answer

Expert verified
(a) The proper time is 12.0 ms. (b) The speed is approximately 99.68% of the speed of light.

Step by step solution

01

Identify Proper Time

Proper time is the time measured by an observer who is at rest relative to the event being measured. In this case, the proper time is the time measured by the first officer on the spacecraft, as the officer is at rest relative to the searchlight.
02

Use Time Dilation Formula

The time dilation formula is given by \( t = \frac{t_0}{\sqrt{1-\frac{v^2}{c^2}}} \), where \( t \) is the time measured by an observer moving relative to the event, \( t_0 \) is the proper time, \( v \) is the velocity of the moving observer, and \( c \) is the speed of light.
03

Convert Measurements to Same Units

Convert the proper time \( t_0 = 12.0 \) ms to seconds: \( 12.0 \text{ ms} = 0.012 \text{ s} \). The observer on Earth measures \( t = 0.150 \text{ s} \).
04

Solve for Velocity Fraction

Rearrange the time dilation formula to solve for \( \frac{v}{c} \): 1. \( t = \frac{t_0}{\sqrt{1-\frac{v^2}{c^2}}} \) 2. Solve for \( \frac{v^2}{c^2} \), giving \( 1-\frac{v^2}{c^2} = \left(\frac{t_0}{t}\right)^2 \)3. \( \frac{v^2}{c^2} = 1-\left(\frac{t_0}{t}\right)^2 \)4. \( \frac{v}{c} = \sqrt{1-\left(\frac{0.012}{0.150}\right)^2} \)
05

Calculate and Simplify

Calculate \( \left(\frac{0.012}{0.150}\right)^2 = 0.0064 \). Therefore, \( \frac{v}{c} = \sqrt{1 - 0.0064} = \sqrt{0.9936} \). This gives \( \frac{v}{c} \approx 0.9968 \).

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Key Concepts

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

Proper Time
In the realm of physics, specifically in the context of special relativity, **proper time** becomes an essential concept. Proper time is the time interval measured by an observer who is at rest relative to the process being observed. It is the true duration of an event from the perspective of someone or something that is not moving relative to the event.

This concept is crucial when dealing with time dilation because the proper time represents the "true" time for events. In our exercise, the officer on the spacecraft measures the searchlight blink duration as 12.0 ms. This is considered the proper time because the officer and the light event are both in the same reference frame—the spacecraft itself. In contrast, the observer on Earth measures a different time, 0.150 s, due to the relativistic effects of time dilation since the spacecraft is moving relative to Earth.
Velocity of Spacecraft
In understanding how fast the spacecraft is moving relative to Earth, we need to delve into the **velocity of the spacecraft** in terms of the speed of light. The formula that governs this relationship is derived from time dilation, which takes into account the effects of an object's speed as it approaches the speed of light.

The important thing to remember is:
  • The observer on Earth perceives a longer time for the searchlight blink due to the relative motion of the spacecraft.
  • This time dilation occurs because the spacecraft is moving at a significant fraction of the speed of light (\( c \)).
  • By rearranging the time dilation formula, we solved for the ratio of the spacecraft's velocity to the speed of light.
In our example, after using the time dilation formula and substituting the known values, we calculated the velocity of the spacecraft as a fraction of the speed of light, approximately 0.9968. This indicates the spacecraft is traveling at almost the speed of light itself, leading to the significant time difference measured by the observers.
Speed of Light
The **speed of light** in a vacuum is a fundamental constant of nature, symbolized by \( c \). It holds a value of approximately 299,792,458 meters per second. This isn't just any random measurement—it defines the ultimate speed limit of the universe.

Key points about the speed of light:
  • Nothing can travel faster than this limit.
  • In our exercise, the speed of light serves as a reference to determine how close the spacecraft's velocity gets to this universal ceiling.
  • The time dilation effects become significant as velocities approach the speed of light, as seen with the spacecraft.
Understanding the speed of light helps in imagining how time can "stretch" and "shrink" depending on the observer's frame of reference and velocity. As the velocity approaches \( c \), time dilation becomes noticeable, which is exactly what we observe in the time difference between the spacecraft and Earth in the exercise.

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Most popular questions from this chapter

Space pilot Mavis zips past Stanley at a constant speed relative to him of 0.800c. Mavis and Stanley start timers at zero when the front of Mavis's ship is directly above Stanley. When Mavis reads 5.00 s on her timer, she turns on a bright light under the front of her spaceship. (a) Use the Lorentz coordinate transformation derived in Example 37.6 to calculate x and t as measured by Stanley for the event of turning on the light. (b) Use the time dilation formula, Eq. (37.6), to calculate the time interval between the two events (the front of the spaceship passing overhead and turning on the light) as measured by Stanley. Compare to the value of \(t\) you calculated in part (a). (c) Multiply the time interval by Mavis's speed, both as measured by Stanley, to calculate the distance she has traveled as measured by him when the light turns on. Compare to the value of \(x\) you calculated in part (a).

(a) Consider the Galilean transformation along the \(x\)-direction : \(x' = x - vt\) and \(t' = t\). In frame \(S\) the wave equation for electromagnetic waves in a vacuum is $${\partial^2E(x, t)\over \partial x^2} - {1 \over c^2} {\partial^2E(x, t)\over \partial t^2} = 0$$ where \(E\) represents the electric field in the wave. Show that by using the Galilean transformation the wave equation in frame \(S'\) is found to be $$(1 - {v^2 \over c^2} ) {\partial^2E(x', t') \over \partial{x'^2}} + {2v \over c^2} {\partial^2E(x', t') \over \partial{x'} \partial{t'}} - {1 \over c^2} {\partial^2E(x', t') \over \partial{t'^2}} = 0$$ This has a different form than the wave equation in \(S\). Hence the Galilean transformation \(violates\) the first relativity postulate that all physical laws have the same form in all inertial reference frames. (\(Hint\): Express the derivatives \(\partial/\partial{x}\) and \(\partial/\partial{t}\) in terms of \(\partial/\partial{x'}\) and \(\partial/\partial{t'}\) by use of the chain rule.) (b) Repeat the analysis of part (a), but use the Lorentz coordinate transformations, Eqs. (37.21), and show that in frame \(S'\) the wave equation has the same form as in frame \(S\): $${\partial^2E(x', t') \over \partial{x'}^2} - {1 \over c^2} {\partial^2E(x', t') \over \partial{t'}^2} = 0$$ Explain why this shows that the speed of light in vacuum is c in both frames \(S\) and \(S'\).

A proton (rest mass 1.67 \(\times\) 10\(^{-27}\) 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?

As you pilot your space utility vehicle at a constant speed toward the moon, a race pilot flies past you in her spaceracer at a constant speed of 0.800c 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\(^8\) 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?

In high-energy physics, new particles can be created by collisions of fast- moving projectile particles with stationary particles. Some of the kinetic energy of the incident particle is used to create the mass of the new particle. A proton-proton collision can result in the creation of a negative kaon (\(K^-\)) and a positive kaon (\(K^+\)): $$p + p \rightarrow p + p + K^- + K^+$$ (a) Calculate the minimum kinetic energy of the incident proton that will allow this reaction to occur if the second (target) proton is initially at rest. The rest energy of each kaon is 493.7 MeV, and the rest energy of each proton is 938.3 MeV. (\(Hint\): It is useful here to work in the frame in which the total momentum is zero. But note that the Lorentz transformation must be used to relate the velocities in the laboratory frame to those in the zero-totalmomentum frame.) (b) How does this calculated minimum kinetic energy compare with the total rest mass energy of the created kaons? (c) Suppose that instead the two protons are both in motion with velocities of equal magnitude and opposite direction. Find the minimum combined kinetic energy of the two protons that will allow the reaction to occur. How does this calculated minimum kinetic energy compare with the total rest mass energy of the created kaons? (This example shows that when colliding beams of particles are used instead of a stationary target, the energy requirements for producing new particles are reduced substantially.)
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