Special relativity question with lots of frames

In summary: Yes, the missile is traveling at speed ##v_M## with respect to the convict ship. The patrol ship sees the missile hit it at time ##t=\frac{v_M}{c^2}##.
  • #1
208
2

Homework Statement


At exactly 00:00:00 hours, a group of convicts escape from a planet in a space-ship that travels at speed
##v=\frac{4}{5}c##.
After 11 min, a patrol spaceship goes after them with ##v_P=\frac{24}{25}c##.
Ignore all acceleration periods.
(i) The convicts immediately notice the patrol spaceship taking off, and release a stealth missile to destroy it. The missile has speed
##v_M=\frac{40}{41}c##
with respect to the convicts. Show that, when the missile hits the patrol ship, the time on a watch worn by the pilot of the patrol ship is 00 : 23 : 36.

Homework Equations


##u_x' = \frac{u_x-v}{1-\frac{u_xv}{c^2}}## (1)
##t' = \gamma (t - \frac{vx}{c^2})## (2)

The Attempt at a Solution


I should say this is from a past paper, so it won't get marked. There are way too many frames! I'm going to assume that all the speeds given are 'proper' speeds.

##u_x=\frac{40}{41}##, I think v is the speed of one frame relative to the other so that might be ##\frac{24}{25}-(-\frac{40}{41}) = \frac{1984}{1025}##. Then I realized that ##u_x'## might not be useful anyway, I was going to just use speed = distance/time, but can't do that because I don't know distances. And (2) probably isn't useful either. Do I try and work out at what time in the ship's frame the missile hits, and transform that? But then do I need to transform the 40/41 to see what the missile's speed is in the ship's frame? Don't I need distances for this approach as well?
 
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  • #2
Kara386 said:
I'm going to assume that all the speeds given are 'proper' speeds.
There is no such thing. It is very well stated in the problem what the speeds are relative to.

What is not clear from the problem is what this means:
Kara386 said:
The convicts immediately notice the patrol spaceship taking off,
Now, there is only one physically sound interpretation of this, but I am not sure that this is what the problem writer intended.

Also, you do not need to consider more than one frame, you simply need to track what happens in one frame and possibly use concepts of relative simultaneity and so on.
 
  • #3
Orodruin said:
There is no such thing. It is very well stated in the problem what the speeds are relative to.

What is not clear from the problem is what this means:

Now, there is only one physically sound interpretation of this, but I am not sure that this is what the problem writer intended.

Also, you do not need to consider more than one frame, you simply need to track what happens in one frame and possibly use concepts of relative simultaneity and so on.
Oops, didn't read it properly, clearly. All speeds relative to the planet. OK, I'll try sticking to the frame of the planet. I thought proper times, length etc were in the rest frame of an object?
 
  • #4
No, all velocities are not given in the planet rest frame, but the velocities in the planet rest frame can be easily figured out using relativistic velocity addition.

Kara386 said:
I thought proper times, length etc were in the rest frame of an object?
Yes, but these are invariant quantities. There is no meaningful way of defining a proper velocity. If you take the velocity of an object in its rest frame it is zero by definition of the rest frame.
 
  • #5
Orodruin said:
No, all velocities are not given in the planet rest frame, but the velocities in the planet rest frame can be easily figured out using relativistic velocity addition.Yes, but these are invariant quantities. There is no meaningful way of defining a proper velocity. If you take the velocity of an object in its rest frame it is zero by definition of the rest frame.
OK. So are the convict ship and spaceship velocities given in the planet's rest frame then? Just the missile one that isn't? If not I really don't understand the question.
 
  • #6
My suggestion would be to calculate the speed of the missile relative to the patrol ship -
at what speed does the patrol ship see the approaching missile?
Then you would need the distance between the two ships after 11 minutes -
this would be relativistically contracted when the patrol ship takes off (you know how far the convict's ship
has traveled relative to earth).
Now if you have this information, speed and distance you should be able
to calculate the time for the missile to reach the patrol ship.
 
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1. What is the concept of "Special Relativity"?

Special Relativity is a theory proposed by Albert Einstein in 1905 to explain the relationship between space and time. It states that the laws of physics are the same for all observers in uniform motion, regardless of their relative speeds or positions. This theory has been proven to accurately describe the behavior of objects moving at high speeds.

2. How does "Special Relativity" differ from "General Relativity"?

Special Relativity deals with the laws of physics in the absence of gravitational forces, while General Relativity takes into account the effects of gravity on the behavior of objects. Special Relativity is based on the principle of relativity and the constancy of the speed of light, while General Relativity incorporates the concept of gravity as a curvature of spacetime.

3. What is the role of frames of reference in "Special Relativity"?

Frames of reference are used in Special Relativity to describe the position and motion of objects in relation to each other. In this theory, all frames of reference are considered equally valid, and the laws of physics should be the same in all frames. This means that observers in different frames will measure different values for quantities such as length, time, and mass, but the laws governing these quantities remain the same.

4. Can "Special Relativity" be applied to everyday situations?

Yes, Special Relativity has been applied to various everyday situations, such as GPS technology, particle accelerators, and satellite communication. It is also used in the design of high-speed vehicles, such as airplanes and spacecraft, to accurately predict their behavior at high velocities.

5. Are there any experiments that have confirmed the predictions of "Special Relativity"?

Yes, there have been many experiments that have confirmed the predictions of Special Relativity. One of the most famous is the Michelson-Morley experiment, which showed that the speed of light is constant regardless of the observer's motion. Other experiments, such as the Hafele-Keating experiment and the Ives-Stilwell experiment, have also confirmed the predictions of Special Relativity.

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