Textbook Question
A cosmic ray travels 60 km through the earth's atmosphere in 400 μs, as measured by experimenters on the ground. How long does the journey take according to the cosmic ray?
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Ch 36: Special Relativity
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796
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Table of contents
- Ch 01: Concepts of Motion35
- Ch 02: Kinematics in One Dimension75
- Ch 03: Vectors and Coordinate Systems58
- Ch 04: Kinematics in Two Dimensions60
- Ch 05: Force and Motion23
- Ch 06: Dynamics I: Motion Along a Line53
- Ch 07: Newton's Third Law27
- Ch 08: Dynamics II: Motion in a Plane52
- Ch 09: Work and Kinetic Energy56
- Ch 10: Interactions and Potential Energy50
- Ch 11: Impulse and Momentum56
- Ch 12: Rotation of a Rigid Body71
- Ch 13: Newton's Theory of Gravity52
- Ch 14: Fluids and Elasticity44
- Ch 15: Oscillations55
- Ch 16: Traveling Waves37
- Ch 17: Superposition39
- Ch 18: A Macroscopic Description of Matter53
- Ch 19: Work, Heat, and the First Law of Thermodynamics60
- Ch 20: The Micro/Macro Connection53
- Ch 21: Heat Engines and Refrigerators34
- Ch 22: Electric Charges and Forces40
- Ch 23: The Electric Field39
- Ch 24: Gauss' Law32
- Ch 25: The Electric Potential63
- Ch 26: Potential and Field57
- Ch 27: Current and Resistance42
- Ch 28: Fundamentals of Circuits39
- Ch 29: The Magnetic Field47
- Ch 30: Electromagnetic Induction60
- Ch 31: Electromagnetic Fields and Waves32
- Ch 32: AC Circuits63
- Ch 33: Wave Optics32
- Ch 34: Ray Optics57
- Ch 35: Optical Instruments30
- Ch 36: Special Relativity38
- Ch 37: The Foundations of Modern Physics25
- Ch 38: Quantization49
- Ch 39: Wave Functions and Uncertainty31
- Ch 40: One-Dimensional Quantum Mechanics35
- Ch 41: Atomic Physics18
- Ch 42: Nuclear Physics27
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook
Chapter 36, Problem 13a
An astronaut travels to a star system 4.5 ly away at a speed of 0.90c. Assume that the time needed to accelerate and decelerate is negligible. How long does the journey take according to Mission Control on earth?
Verified step by step guidance1
Step 1: Identify the given values in the problem. The distance to the star system is 4.5 light-years (ly), and the speed of the astronaut is 0.90c, where c is the speed of light (approximately 3.00 × 10⁸ m/s).
Step 2: Recall the formula for time, which is derived from the relationship between distance, speed, and time: , where is the time, is the distance, and is the velocity.
Step 3: Substitute the given values into the formula. The distance is 4.5 ly, and the velocity is 0.90c. The formula becomes: (in light-years per year).
Step 4: Simplify the fraction to determine the time in years. Note that the units of light-years and years are consistent, so no unit conversion is necessary.
Step 5: Conclude that the time calculated represents the duration of the journey as observed by Mission Control on Earth, since the calculation is based on the distance and speed in the Earth reference frame.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Light-Year (ly)
A light-year is a unit of distance that represents how far light travels in one year. Since light moves at approximately 299,792 kilometers per second, one light-year is about 9.46 trillion kilometers. This concept is crucial for understanding astronomical distances and helps quantify the vastness of space, such as the 4.5 light-years to the star system in the question.
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Relativistic Speed
Relativistic speed refers to velocities that are a significant fraction of the speed of light (denoted as 'c'). At these speeds, the effects of Einstein's theory of relativity become significant, particularly time dilation and length contraction. In this scenario, the astronaut's speed of 0.90c means that relativistic effects must be considered when calculating the journey's duration from different reference frames.
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Time Dilation
Time dilation is a phenomenon predicted by Einstein's theory of relativity, where time passes at different rates for observers in different frames of reference. For an observer moving at a relativistic speed, time appears to pass more slowly compared to an observer at rest. This concept is essential for understanding how the journey duration is perceived by Mission Control on Earth versus the astronaut traveling at 0.90c.
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Related Practice
Textbook Question
You fly 5000 km across the United States on an airliner at 250 m/s. You return two days later at the same speed. Have you aged more or less than your friends at home?
Textbook Question
Your job is to synchronize the clocks in a reference frame. You are going to do so by flashing a light at the origin at t = 0 s. To what time should the clock at (x, y, z) = (30 m, 40 m, 0 m) be preset?
Textbook Question
A baseball pitcher can throw a ball with a speed of 40 m/s. He is in the back of a pickup truck that is driving away from you. He throws the ball in your direction, and it floats toward you at a lazy 10 m/s. What is the speed of the truck?
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Textbook Question
Bjorn is standing at x = 600 m. Firecracker 1 explodes at the origin and firecracker 2 explodes at x = 900 m. The flashes from both explosions reach Bjorn's eye at t = 3.0 μs. At what time did each firecracker explode?
Textbook Question
You fly 5000 km across the United States on an airliner at 250 m/s. You return two days later at the same speed. By how much? Hint: Use the binomial approximation.