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 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 1.20×1081.20\$$times10^8 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?

Ch 37: Special Relativity

Young & Freedman Calc - University Physics 14th Edition

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Young & Freedman Calc 14th Edition

Ch 37: Special Relativity

Problem 6

Chapter 37, Problem 6

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 $0.800 c$ 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 $1.20 \times 10^{8}$ 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?

Verified step by step guidance

Step 1: Understand the problem involves relative motion at relativistic speeds, so we need to use the principles of special relativity, particularly time dilation and length contraction.

Step 2: For part (a), use the time dilation formula: $ t' = \frac{t}{\sqrt{1 - \frac{v^2}{c^2}}} $, where $ t' $ is the time measured by the race pilot, $ t $ is the time measured by you, $ v = 0.800c $ is the relative speed, and $ c $ is the speed of light. Calculate $ t $ using the distance $ 1.20 \times 10^8 $ m and speed $ 0.800c $.

Step 3: For part (b), use the concept of length contraction. The distance you measure is contracted in the race pilot's frame. Use the formula: $ L' = L \sqrt{1 - \frac{v^2}{c^2}} $, where $ L $ is the distance you measure and $ L' $ is the distance measured by the race pilot.

Step 4: For part (c), use the time dilation formula again to find the time on your timer when the race pilot's timer reads the value calculated in part (a). Rearrange the time dilation formula to solve for $ t $ in terms of $ t' $.

Step 5: Ensure all calculations respect the relativistic effects due to the high speed (0.800c) and verify each step with the appropriate relativistic formulas to ensure consistency in the results.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Relativity of Simultaneity

The relativity of simultaneity is a principle in Einstein's theory of relativity stating that events perceived as simultaneous in one frame of reference may not be simultaneous in another. This concept is crucial for understanding how different observers measure time and distance differently when moving at high speeds, such as 0.800c, relative to each other.

Time Dilation

Time dilation is a phenomenon predicted by the theory of relativity, where time is observed to pass at different rates for observers in different inertial frames, especially at high velocities close to the speed of light. This concept helps explain why the race pilot's timer reads differently from yours, as her high speed affects her perception of time compared to yours.

Length Contraction

Length contraction is another relativistic effect where the length of an object moving at a significant fraction of the speed of light appears shorter along the direction of motion to a stationary observer. This concept is essential for understanding how the race pilot measures your distance differently, as her high speed alters her perception of spatial dimensions compared to yours.

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Length Contraction

Related Practice

Textbook Question

Why Are We Bombarded by Muons? Muons are unstable subatomic particles that decay to electrons with a mean lifetime of 2.2 μs. They are produced when cosmic rays bombard the upper atmosphere about 10 km above the earth's surface, and they travel very close to the speed of light. The problem we want to address is why we see any of them at the earth's surface. (a) What is the greatest distance a muon could travel during its 2.2 μs lifetime? (b) According to your answer in part (a), it would seem that muons could never make it to the ground. But the 2.2 μs lifetime is measured in the frame of the muon, and muons are moving very fast. At a speed of 0.999c, what is the mean lifetime of a muon as measured by an observer at rest on the earth? How far would the muon travel in this time? Does this result explain why we find muons in cosmic rays? (c) From the point of view of the muon, it still lives for only 2.2 μs, so how does it make it to the ground? What is the thickness of the 10 km of atmosphere through which the muon must travel, as measured by the muon? Is it now clear how the muon is able to reach the ground?

Textbook Question

An unstable particle is created in the upper atmosphere from a cosmic ray and travels straight down toward the surface of the earth with a speed of 0.99540c relative to the earth. A scientist at rest on the earth’s surface measures that the particle is created at an altitude of 45.0 km. (a) As measured by the scientist, how much time does it take the particle to travel the 45.0 km to the surface of the earth? (b) Use the length-contraction formula to calculate the distance from where the particle is created to the surface of the earth as measured in the particle’s frame. (c) In the particle’s frame, how much time does it take the particle to travel from where it is created to the surface of the earth? Calculate this time both by the time dilation formula and from the distance calculated in part (b). Do the two results agree?

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Textbook Question

Suppose the two lightning bolts shown in Fig. 37.5a are simultaneous to an observer on the train. Show that they are not simultaneous to an observer on the ground. Which lightning strike does the ground observer measure to come first?

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Textbook Question

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 $c$ ?

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Textbook Question

The positive muon (µ⁺), an unstable particle, lives on average 2.20 × 10^-6 s (measured in its own frame of reference) before decaying. If such a particle is moving, with respect to the laboratory, with a speed of 0.900c, what average lifetime is measured in the laboratory?

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Textbook Question

The positive muon (µ⁺), an unstable particle, lives on average 2.20 × 10^-6 s (measured in its own frame of reference) before decaying. What average distance, measured in the laboratory, does the particle move before decaying?

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