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0. Math Review31m
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1. Intro to Physics Units1h 29m
2. 1D Motion / Kinematics3h 56m
3. Vectors2h 43m
4. 2D Kinematics1h 42m
5. Projectile Motion3h 6m
6. Intro to Forces (Dynamics)3h 22m
7. Friction, Inclines, Systems2h 44m
8. Centripetal Forces & Gravitation7h 26m
9. Work & Energy1h 59m
10. Conservation of Energy2h 54m
11. Momentum & Impulse3h 40m
12. Rotational Kinematics2h 59m
13. Rotational Inertia & Energy7h 4m
14. Torque & Rotational Dynamics2h 5m
15. Rotational Equilibrium3h 39m
16. Angular Momentum3h 6m
17. Periodic Motion2h 9m
18. Waves & Sound3h 40m
19. Fluid Mechanics4h 27m
20. Heat and Temperature3h 7m
21. Kinetic Theory of Ideal Gases1h 50m
22. The First Law of Thermodynamics1h 26m
23. The Second Law of Thermodynamics3h 11m
24. Electric Force & Field; Gauss' Law3h 42m
25. Electric Potential1h 51m
26. Capacitors & Dielectrics2h 2m
27. Resistors & DC Circuits3h 8m
28. Magnetic Fields and Forces2h 23m
29. Sources of Magnetic Field2h 30m
30. Induction and Inductance3h 38m
31. Alternating Current2h 37m
32. Electromagnetic Waves2h 14m
33. Geometric Optics2h 57m
34. Wave Optics1h 15m
35. Special Relativity2h 10m

35. Special Relativity

Inertial Reference Frames

Physics

35. Special Relativity

Inertial Reference Frames

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Problem 13a

Textbook Question

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 guidance

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:

t = \frac{d}{v}

, where

t

is the time,

d

is the distance, and

v

is the velocity.

Step 3: Substitute the given values into the formula. The distance

d

is 4.5 ly, and the velocity

v

is 0.90c. The formula becomes:

t = \frac{4.5}{0.90}

(in light-years per year).

Step 4: Simplify the fraction to determine the time

t

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.

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.

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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Textbook Question

Jill claims that her new rocket is 100 m long. As she flies past your house, you measure the rocket’s length and find that it is only 80 m. What is Jill’s speed, as a fraction of c?

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A cube has a density of 2000 kg/m³ while at rest in the laboratory. What is the cube’s density as measured by an experimenter in the laboratory as the cube moves through the laboratory at 90% of the speed of light in a direction perpendicular to one of its faces?

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The Stanford Linear Accelerator (SLAC) accelerates electrons to v = 0.99999997c in a 3.2-km-long tube. If they travel the length of the tube at full speed (they don’t, because they are accelerating), how long is the tube in the electrons’ reference frame?

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A rocket traveling at 0.50c sets out for the nearest star, Alpha Centauri, which is 4.3 ly away from earth. It will return to earth immediately after reaching Alpha Centauri. What distance will the rocket travel and how long will the journey last according to (a) stay-at-home earthlings and (b) the rocket crew? (c) Which answers are the correct ones, those in part a or those in part b?

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Two events in reference frame S occur 10 μs apart at the same point in space. The distance between the two events is 2400 m in reference frame S'. What is the velocity of S' relative to S?

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In an attempt to reduce the extraordinarily long travel times for voyaging to distant stars, some people have suggested traveling at close to the speed of light. Suppose you wish to visit the red giant star Betelgeuse, which is 430 ly away, and that you want your 20,000 kg rocket to move so fast that you age only 20 years during the round trip. How fast, as a fraction of c, must the rocket travel relative to earth?

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