Velocity in Special Relativity

In summary, to travel to Pleiades in 10 years according to your clock, you would need to travel at a velocity of 0.9997 times the speed of light. When you reach Pleiades and send a radio signal back to Earth, for someone on Earth, it would have been 408.3 years since you left (assuming the distance of 130 pc was measured in the rest frame).
  • #1
Kamkazemoose
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Homework Statement


You wish to travel to Pleisdes (at a distance of 130 pc) in 10 years, according to the clock that you carry. How fast do you have to travel to accomplish this (express the velocity as v/c)?
When you reach the Pleisdes, you send a radio signal back to Earth. For someone on Earth, how long has it been between the moment you left and the moment when the radio signal was received?


Homework Equations


t=[tex]\gamma[/tex]t0
d=[tex]\gamma[/tex]d0
[tex]\gamma[/tex]=[tex]\frac{1}{1-\sqrt{1-\frac{v^2}{c^2}}}[/tex]
v=d/t

The Attempt at a Solution


I know that to calculate velocity, you have to use the time and distance in the same inertial frame, so we can't simply insert the given d and the given t, as they are in 2 frames. I tried to come up with an equation for velocityby converting t0 to t, so I had
(using t(naught) as t0, because i can't get it to work within latex, sorry)

v=[tex]\frac{d}{\gamma*t(naught)}[/tex]

and then, replacing gamma and using algebra etc, I simplified this to

v=[tex]\frac{d}{\sqrt{t(naught)^2+\frac{d^2}{c^2}}}[/tex]

where t0 = 10yr and d= 130 pc
I got that v=2.9996*108m/s or v=.9997c and [tex]\gamma[/tex] = 40.83
then, when I calculate t in the frame of the earth, i get t=408.3 years, but d = 424lightyears, so This should mean that the person is traveling faster than the speed of light, but I found v as less than the speed of light, so something went wrong.

If you could help, did I get the equation for finding velocity wrong, or was there something else, I can't figure it out, thanks. Also, sorry but I don't quite understand latex, so there may be somethings that look ugly.
 
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  • #2
10 light years is the time length according to the moving frame, but what about the distance is that the distance you percieve in the moving frame or is that the distance from the rest frame, according to an observer on Earth?!

If 130pc is the length observed in the moving frame then calculating the velocity is easy because all quantities are in the same frame of reference. If the distance is relative to a 'stationary' observer in the rest frame of the Earth, remember 'moving clocks run slow' so the distance should be a smaller number (just to help as a self check due to length contraction)

The second part would follow as such

Time = time it took you get there(10*gamma) + time it took signal to travel distance of 130pc (if 130 pc was measured in the rest frame, if not then adjust the distance anf find the tiem it took, this is easy since your measuring the distance in light years)
 
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Related to Velocity in Special Relativity

1. What is the definition of velocity in special relativity?

Velocity in special relativity is a measure of an object's speed and direction of motion as observed from a particular frame of reference. It takes into account the effects of time dilation and length contraction due to the relative motion between the observer and the object.

2. How is velocity calculated in special relativity?

In special relativity, the formula for calculating velocity is v = u / (1 - (u^2/c^2)), where v is the velocity of the object in the observer's frame of reference, u is the velocity of the object in its own frame of reference, and c is the speed of light.

3. Can an object in special relativity travel faster than the speed of light?

No, according to the theory of special relativity, the speed of light is the maximum speed at which any object can travel. As an object approaches the speed of light, its mass increases and it requires an infinite amount of energy to accelerate it further. Therefore, it is impossible for an object to travel faster than the speed of light.

4. How does velocity in special relativity affect time?

According to the theory of special relativity, time is relative and is affected by an object's velocity. As an object's velocity increases, time for that object appears to slow down for an observer in a different frame of reference. This is known as time dilation.

5. What is the difference between velocity in special relativity and classical mechanics?

In classical mechanics, velocity is calculated as v = d/t, where d is the distance traveled and t is the time taken. This formula assumes that time is independent of the observer's frame of reference, which is not the case in special relativity. In special relativity, the concept of velocity is more complex and takes into account the effects of time dilation and length contraction due to the relative motion between the observer and the object.

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