# Show the relative difference in clock time between plane/ground is

• akennedy
In summary, the homework statement asks for a solution to a problem that is dependent on two unknowns, the angular frequency of the Earth's rotation and the plane velocity of the object being rotated. The student attempted to solve the problem by deriving the equation from the knowns, but they had to make an approximation for the negligible terms.
akennedy

## Homework Statement

http://puu.sh/7vC46.png
http://puu.sh/7vBG5.png
Note: In this image the ohm symbol represents the angular frequency of the Earth's rotation

## Homework Equations

Gamma = 1/SQRT(1-v^2/c^2)
V (relative to centre of earth) = angular frequency(R+h) +/- v
v= plane velocity dependent on direction
h= height plane is flying

## The Attempt at a Solution

The first part about showing it isn't dependent was easy and I just did some basic algebra. However, the second part is problematic.I can't seem to get to the equation it's asking for using the 2 tricks suggested. I think possibly because I'm using the wrong velocities when putting in Gamma(earth) and Gamma(plane)

The velocities I used were
V(plane) = Angular frequency*(R+h)+/-v
V(Earth) = Angular frequency*(R+h)
I put those into the gamma formula and went from there. I think I might be using the wrong velocities and have mucked up the inertial frames or something.

Last edited by a moderator:
For what it's worth, I was able to derive the equation using
V(Earth) = Angular frequency*R

I figured since that clock is on the Earth, there's no reason to include "h" for that particular clock (The "h" is still present in the planes' velocities though).

But, and this is a big "but," I also had to assume that $\frac{\Omega^2 h^2}{c^2}$ and $\frac{\Omega h v}{c^2}$ were negligible compared to other, similar terms and can be neglected (contrasted with terms like $\frac{R \Omega^2 h}{c^2}$ and $\frac{R \Omega v}{c^2}$ which are obviously much larger). That particular approximation wasn't spelled out in the problem statement, so I'm not sure if my method is "correct" either.

1 person
Omg... Wow thank you so much. I can't believe I overlooked that haha :P

## 1. What is the relative difference in clock time between a plane and the ground?

The relative difference in clock time between a plane and the ground is known as time dilation. This is due to the fact that time moves slower for objects in motion compared to those at rest. As a plane travels at high speeds, it experiences time dilation, causing its clock to run slower compared to a clock on the ground.

## 2. How is the relative difference in clock time between a plane and the ground measured?

The relative difference in clock time between a plane and the ground can be measured using the formula t' = t√(1-v^2/c^2), where t' is the time measured on the moving clock, t is the time measured on the stationary clock, v is the velocity of the moving clock, and c is the speed of light.

## 3. Does the relative difference in clock time between a plane and the ground affect GPS systems?

Yes, the relative difference in clock time between a plane and the ground does affect GPS systems. The atomic clocks used in GPS satellites are adjusted to account for time dilation caused by their high speeds in orbit. Without this adjustment, GPS systems would be inaccurate by several kilometers.

## 4. Is the relative difference in clock time between a plane and the ground significant?

The relative difference in clock time between a plane and the ground is relatively small, but it can become significant over long distances and high speeds. For example, a plane traveling at 600 mph would experience a time dilation of about 7 microseconds per day compared to a clock on the ground.

## 5. Can the relative difference in clock time between a plane and the ground be observed?

Yes, the relative difference in clock time between a plane and the ground can be observed using precision atomic clocks. However, the effects are so small that they would not be noticeable to the human eye. Only highly accurate and sensitive measurements can detect the difference in clock time between a plane and the ground.

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