Calculating Time Dilation: Apollo's Record-Breaking Speed and Earth's Clocks

In summary: So, the time dilation for the Apollo spacecraft is about 9.985%.In summary, the record for the fastest speed ever traveled relative to the Earth is held by the Apollo X module at 24,791 mi/h. When calculating the time difference between the clocks on the Apollo and Earth, using the Lorentz transformation, the time dilation for the Apollo spacecraft is about 9.985%, which equates to approximately 6.82x10^-8 s. The discrepancy between this and the given answer of 6.82x10^-8 s may be due to a miscalculation in the square root calculation.
  • #1
Quelsita
49
0
Question:

The record for the fastest speed at which anyone has ever traveled, relative to the Earth, is held by the Apollo X modue at 24,791 mi/h on their return trip from the moon.
At this speed, what is the percent difference between the clocks on the Apollo and the clocks on Earth?

OK, I think I understand how to do the problem, but I'm not getting the answer that our text gives.

I set the moving IRF as the Apollo and the stationary frame as the Earth.
I then found the time elapsed in the frame of the spaceship by using d=vt and the given velocity and the distance from the Earth to the moon.
So:

VApollo=V=24,791mi/h=11082.3m/s
d(Earth to Moon)=3.84x10^8 m

With no time dilation for the ship, the time elapsed then is:
from d=vt: tApollo=t'=3.84x10^8 m/11082.3m/s = 34649.85 s

I then found the time dilation for the time elapsed in Earth's reference frame using Lorentz tranformation:
t=t'/[tex]\sqrt{1-(V/c)^2}[/tex]

gives an elapsed time in Earth' IRF of 346449.85s

and 346449.85s/34649.85s = 9.985%

The correct answer is 6.82x10^-8 s

Could someone help me figure out where I went wrong? I went over the math a few times, so it must be my logic...

Thanks!
 
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  • #2
I don't have a calculator handy, but something went wrong in that square root calc. Did you note that the spacecraft speed is in mph, so in the same units c = 186000*3600 ?

Doubtless that square root is well within 1% of 1.
 
  • #3
Yeah, I actually converted the velocity to m/s I just skipped the conversion work.
 

1. What is time dilation?

Time dilation is a phenomenon in which time appears to pass at different rates for objects in different frames of reference. It is a consequence of Einstein's theory of relativity and is caused by the effects of gravity and motion on the fabric of space-time.

2. How does time dilation occur?

Time dilation occurs due to the fact that the speed of light is constant in all frames of reference. As an object moves closer to the speed of light, time appears to slow down for that object relative to an observer in a different frame of reference. Similarly, objects in stronger gravitational fields experience time dilation due to the warping of space-time.

3. What is the equation for time dilation?

The equation for time dilation is t' = t / √(1-(v^2/c^2)), where t' is the time measured in the frame of reference of the moving object, t is the time measured by an observer in a stationary frame of reference, v is the velocity of the moving object, and c is the speed of light.

4. Can time dilation be observed?

Yes, time dilation has been observed and measured in experiments with atomic clocks in airplanes and satellites. These clocks experience slightly slower time due to their high speeds and distance from the Earth's gravitational field.

5. What are the practical implications of time dilation?

Time dilation has significant implications for space travel and GPS technology. For astronauts traveling at high speeds, time dilation can cause them to age slower than people on Earth. This must be taken into account for long space missions. For GPS technology, time dilation must be accounted for to ensure accurate location data as the satellites are moving at high speeds relative to the Earth's surface.

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