The relativity of time: Time dilation

[MrDMD83]

Homework Statement

Suppose that you are traveling on board a spacecraft that is moving with respect to the Earth at a speed of 0.925c. You are breathing at a rate of 10.0 breaths per minute. As monitored on earth, what is your breathing rate in breaths per minute?

Homework Equations

t=t0/(1-v^2/c^2)^1/2

The Attempt at a Solution

t=1/(1-(.925x3e8)^2/(3e8)^2)^1/2

Not really sure where to go from here.

 

[Doc Al]

You are on the right track--don't stop now.

Your breathing is a "clock" like any other. Viewed from the earth, you are a moving clock which is measured to run slowly by a factor given by that equation. Complete the calculation!

Note: Better to write the equation for "time dilation" as:
$$\Delta T = \Delta T_0 \frac{1}{\sqrt{1 - v^2/c^2}}$$

Where $$\Delta T_0$$ is the time it takes you to breath 10 breaths (1 minute, as measured by you) and $$\Delta T$$ is the time (the number of Earth minutes) that Earth measurements say it took you to breath those 10 breaths. Use that Earth time to calculate your breathing rate according to Earth clocks.

 

[m2003]

The concept of time dilation is a fundamental principle in Einstein's theory of relativity, which states that the passage of time is not absolute and can vary depending on the relative motion between two observers. In this scenario, as you are traveling at a speed of 0.925c, which is close to the speed of light, the time dilation effect will be significant. This means that time will appear to pass slower for you on the spacecraft compared to an observer on Earth.

To calculate your breathing rate as monitored on Earth, we can use the equation t=t0/(1-v^2/c^2)^1/2, where t0 is the breathing rate as measured by you on the spacecraft and t is the breathing rate as measured on Earth. Plugging in the values, we get t=10/(1-(0.925c)^2/c^2)^1/2 = 10/(1-0.854)^1/2 = 10/(0.146)^1/2 = 68.5 breaths per minute.

This means that your breathing rate as measured on Earth will be significantly higher (almost 7 times) than your actual breathing rate on the spacecraft. This is due to the time dilation effect caused by your high speed relative to Earth. This concept of time dilation has been tested and confirmed through various experiments and is an important aspect of understanding the nature of time and space in our universe.