Ch 36: Special Relativity

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 36: Special Relativity

Problem 14

Chapter 36, Problem 14

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?

Verified step by step guidance

Step 1: Identify the concept involved in the problem. This is a relativistic problem involving time dilation, which occurs when an object moves at a significant fraction of the speed of light. The time experienced by the cosmic ray (proper time) is shorter than the time measured by observers on the ground due to its high velocity.

Step 2: Write down the formula for time dilation:

t = t

₀/1-v²/c², where

t

is the time measured by the ground observers,

t

₀ is the proper time experienced by the cosmic ray,

v

is the velocity of the cosmic ray, and

c

is the speed of light.

Step 3: Rearrange the formula to solve for the proper time

t

₀:

t

₀=t×1-v²/c². Substitute the given values:

t

= 400 μs, and calculate

v

using the distance traveled (60 km) and the time measured by the ground observers.

Step 4: Calculate the velocity of the cosmic ray:

v = d / t

, where

d

is the distance (60 km converted to meters) and

t

is the time (400 μs converted to seconds). Ensure the units are consistent before performing the calculation.

Step 5: Substitute the calculated velocity

v

and the speed of light

c

(approximately 3 × 10⁸ m/s) into the time dilation formula to find the proper time

t

₀. This will give the time experienced by the cosmic ray during its journey.

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Key Concepts

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

Time Dilation

Time dilation is a phenomenon predicted by Einstein's theory of relativity, where time is observed to pass at different rates for observers in different frames of reference. For objects moving at speeds close to the speed of light, such as cosmic rays, time appears to pass more slowly compared to stationary observers. This concept is crucial for understanding how the cosmic ray perceives its travel time compared to the time measured by experimenters on the ground.

Lorentz Factor

The Lorentz factor is a mathematical factor that arises in the equations of special relativity, defined as γ = 1 / √(1 - v²/c²), where v is the velocity of the moving object and c is the speed of light. This factor quantifies the effects of time dilation and length contraction experienced by objects moving at relativistic speeds. It is essential for calculating how much time elapses for the cosmic ray compared to the time measured by stationary observers.

Relativistic Speeds

Relativistic speeds refer to velocities that are a significant fraction of the speed of light (approximately 3 x 10^8 m/s). At these speeds, classical mechanics no longer accurately describes motion, and relativistic effects, such as time dilation and length contraction, become significant. Understanding these speeds is vital for analyzing the behavior of cosmic rays as they travel through the atmosphere and how their experiences differ from those of observers on Earth.

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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

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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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.