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insynC
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I've got an examination in special relativity in a couple of days and I've got four questions that have been bugging me. I thought it would be better to lump them all together as they're all on the same topic, but sorry if it's a big read! If you reckon I'd be better off posting them as four questions, let me know and I'll fix things up. Any help you could give me with any of the questions would be greatly appreciated.
Question 1
The proper time between flashes of a lighthouse is 2.0 secs. What is the the measured interval between flashes for an observer traveling at 0.4c with respect to the lighthouse?
This one has been bugging me the most, every time I think I'm on top of time dilation it gets me again. I'm not particularly interested in the numerical solution, more on whether the 'moving' observer will see a period of greater or less than 2.0 secs. In my class my lecturer said it will be less than 2.0 secs, but I'm not convinced.
I am able to convince myself just about anything with t = \gamma \tau, so I'll set out what I'm thinking in terms of a more heuristic (aka hand waving) argument:
There is something in the lighthouse that determines when 2.0 secs has passed to let out another light pulse, so let that be a light clock (photon bouncing between two walls, each bounce is a tick = unit of time). With respect to the spaceship the lighthouse and consequently the light clock is moving. Thus the photon has to travel further between each 'tick' and so each tick in the lighthouse frame is slower than the proper time for the observer. Thus for 2.0 seconds worth of ticks to pass in the lighthouse frame now will take longer than 2.0 secs in the observer frame. Thus I expect the period to be greater than 2.0 secs.
Just to check, in the lighthouse frame of reference the pulses are let out every 2.0 seconds as normal. But the lighthouse sees time in the passing observer frame to be slower (by same light clock argument as above) thus it will expect the observer to measure greater than 2.0 seconds for the time in between pulses.
I realize this is a really fundamental SR question, but this course is my first introduction to the topic. Am I right in my argument, or was my lecturer right all along? If the latter, what is leading me astray in my reasoning?
Question 2
Derive the Lorentz transformation for the x component of momentum, i.e.
Px' = \gamma (Px - vE/(c^{}2))
I've used Px = x component of momentum (not very good with latex, sorry!)
I thought the best place to start was the Lorentz transformation for velocity (which was given):
ux' = [ux - v] / [1 + v ux/(c^{}2)]
Applying this, I used the fact Px = \gamma m0 ux - where m0 is rest mass - and then fiddled around with it.
I was able to almost get the answer, except on the RHS I got what is required multiplied by a factor of:
1 / [ \gamma - \gamma ux v /(c^{}2) ]
Unfortunately I couldn't show this was equal to 1 and am not even convinced it is. Was the approach I took the easiest way to the answer? I've tried it again and got the same problem, so maybe there is a better way to tackle it.
Question 3
A collision between two protons can result in the creation of a positive pion and the conversion of one proton to a neutron:
p^{}+ + p^{}+ --> p^{}+ + n + \pi ^{}+
(The last one is a positive pion, again sorry about my bad use of latex.)
Calculate the minimum kinetic energy (in MeV) for the protons in this reaction if the two protons have equal energy.
I think conservation of energy and momentum are the key to solving this question.
The fact the two initial protons means that as they have the same rest mass, they will have the same momentum and so the momentum of the initial system, and hence the final system, must be zero.
Thus while maintaining the total momentum as zero, I know I have to adjust the velocities of the three final particles to minimise the total energy of the system.
As momentum is proportional to v \gamma and energy is proportional to \gamma, my thought is that the gamma factor for the more massive particles (neutron and proton) need to be minimised whilst the gamma for the pion needs to be maximised, as conceptually this should provide the minimum energy whilst still conserving momentum.
Nonetheless actually putting this into action has not led me to any success. I'm not sure if this is the right way to approach the problem, but I have the proton and neutron heading off perpendicularly (say in an x-y plane the proton in the -x direction and the neutron in the -y direction) whilst the pion is at some angle in the first quadrant (where x & y are positive).
Trying to solve the equations though are not only horrendous, but I end up with two variables in the one equation: \theta (angle pion makes with x axis) and the gamma factor for the pion.
Is there a better way to approach this problem?
Question 4
A cosmonaut spends a few years in an orbit above the Earth. We would like to estimate how his age will differ from his age if he had stayed on Earth. We will consider two separate effects.
(a) First calculate the effect due to time dilation from Special Relativity. Let the cosmonaut be orbiting in a circular orbit at a height 200 km above the Earth's surface. Assume that the velocity at the Earth's surface is negligible. What is the ratio of the cosmonaut's time interval compared to the time interval at the Earth's surface?
(b) The second effect is due to gravitational redshift. Write down an expression for the ratio between the time intervals at the surface of the Earth and in the cosmonaut's spaceship. What is the value of this ratio for the values given in the previous part of the question?
(c) In part (a) we assumed that the velocity at the Earth's surface was negligible. Explain why this is a reasonable assumption.
[You may take the radius of the Earth to be 6380km.]
Doppler shift equation:
\lambda1 / \lambda2 = 1 + z = sqrt[(1 +v/c)/(1-v/c)]
Gravitational redshift:
\lambda / \lambda0 = 1 + z = [1 - 2GM/(c^{}2R)]^(-1/2)
I know time is proportional to 1/frequency, so I'm going to need to use the Doppler shift equation in part (a) and the Gravitational redshift equation for part (b).
The fact I am given a \DeltaR (200km) as the distance above the Earth makes me think I'm going to need to apply calculus to these equations. But I'm not exactly sure how to approach this
For (a) I think I might need to use the radial velocity equation to determine v, then perhaps differentiate this. But I am not sure whether this is the right approach, and even if it is how to go about it.
I am not sure at all about (c)
Conclusion
Again many apologies for such a long post and many thanks to anyone who can help me with any of these problems.
Question 1
Homework Statement
The proper time between flashes of a lighthouse is 2.0 secs. What is the the measured interval between flashes for an observer traveling at 0.4c with respect to the lighthouse?
The Attempt at a Solution
This one has been bugging me the most, every time I think I'm on top of time dilation it gets me again. I'm not particularly interested in the numerical solution, more on whether the 'moving' observer will see a period of greater or less than 2.0 secs. In my class my lecturer said it will be less than 2.0 secs, but I'm not convinced.
I am able to convince myself just about anything with t = \gamma \tau, so I'll set out what I'm thinking in terms of a more heuristic (aka hand waving) argument:
There is something in the lighthouse that determines when 2.0 secs has passed to let out another light pulse, so let that be a light clock (photon bouncing between two walls, each bounce is a tick = unit of time). With respect to the spaceship the lighthouse and consequently the light clock is moving. Thus the photon has to travel further between each 'tick' and so each tick in the lighthouse frame is slower than the proper time for the observer. Thus for 2.0 seconds worth of ticks to pass in the lighthouse frame now will take longer than 2.0 secs in the observer frame. Thus I expect the period to be greater than 2.0 secs.
Just to check, in the lighthouse frame of reference the pulses are let out every 2.0 seconds as normal. But the lighthouse sees time in the passing observer frame to be slower (by same light clock argument as above) thus it will expect the observer to measure greater than 2.0 seconds for the time in between pulses.
I realize this is a really fundamental SR question, but this course is my first introduction to the topic. Am I right in my argument, or was my lecturer right all along? If the latter, what is leading me astray in my reasoning?
Question 2
Homework Statement
Derive the Lorentz transformation for the x component of momentum, i.e.
Px' = \gamma (Px - vE/(c^{}2))
I've used Px = x component of momentum (not very good with latex, sorry!)
Homework Equations
I thought the best place to start was the Lorentz transformation for velocity (which was given):
ux' = [ux - v] / [1 + v ux/(c^{}2)]
The Attempt at a Solution
Applying this, I used the fact Px = \gamma m0 ux - where m0 is rest mass - and then fiddled around with it.
I was able to almost get the answer, except on the RHS I got what is required multiplied by a factor of:
1 / [ \gamma - \gamma ux v /(c^{}2) ]
Unfortunately I couldn't show this was equal to 1 and am not even convinced it is. Was the approach I took the easiest way to the answer? I've tried it again and got the same problem, so maybe there is a better way to tackle it.
Question 3
Homework Statement
A collision between two protons can result in the creation of a positive pion and the conversion of one proton to a neutron:
p^{}+ + p^{}+ --> p^{}+ + n + \pi ^{}+
(The last one is a positive pion, again sorry about my bad use of latex.)
Calculate the minimum kinetic energy (in MeV) for the protons in this reaction if the two protons have equal energy.
Homework Equations
I think conservation of energy and momentum are the key to solving this question.
The Attempt at a Solution
The fact the two initial protons means that as they have the same rest mass, they will have the same momentum and so the momentum of the initial system, and hence the final system, must be zero.
Thus while maintaining the total momentum as zero, I know I have to adjust the velocities of the three final particles to minimise the total energy of the system.
As momentum is proportional to v \gamma and energy is proportional to \gamma, my thought is that the gamma factor for the more massive particles (neutron and proton) need to be minimised whilst the gamma for the pion needs to be maximised, as conceptually this should provide the minimum energy whilst still conserving momentum.
Nonetheless actually putting this into action has not led me to any success. I'm not sure if this is the right way to approach the problem, but I have the proton and neutron heading off perpendicularly (say in an x-y plane the proton in the -x direction and the neutron in the -y direction) whilst the pion is at some angle in the first quadrant (where x & y are positive).
Trying to solve the equations though are not only horrendous, but I end up with two variables in the one equation: \theta (angle pion makes with x axis) and the gamma factor for the pion.
Is there a better way to approach this problem?
Question 4
Homework Statement
A cosmonaut spends a few years in an orbit above the Earth. We would like to estimate how his age will differ from his age if he had stayed on Earth. We will consider two separate effects.
(a) First calculate the effect due to time dilation from Special Relativity. Let the cosmonaut be orbiting in a circular orbit at a height 200 km above the Earth's surface. Assume that the velocity at the Earth's surface is negligible. What is the ratio of the cosmonaut's time interval compared to the time interval at the Earth's surface?
(b) The second effect is due to gravitational redshift. Write down an expression for the ratio between the time intervals at the surface of the Earth and in the cosmonaut's spaceship. What is the value of this ratio for the values given in the previous part of the question?
(c) In part (a) we assumed that the velocity at the Earth's surface was negligible. Explain why this is a reasonable assumption.
[You may take the radius of the Earth to be 6380km.]
Homework Equations
Doppler shift equation:
\lambda1 / \lambda2 = 1 + z = sqrt[(1 +v/c)/(1-v/c)]
Gravitational redshift:
\lambda / \lambda0 = 1 + z = [1 - 2GM/(c^{}2R)]^(-1/2)
The Attempt at a Solution
I know time is proportional to 1/frequency, so I'm going to need to use the Doppler shift equation in part (a) and the Gravitational redshift equation for part (b).
The fact I am given a \DeltaR (200km) as the distance above the Earth makes me think I'm going to need to apply calculus to these equations. But I'm not exactly sure how to approach this
For (a) I think I might need to use the radial velocity equation to determine v, then perhaps differentiate this. But I am not sure whether this is the right approach, and even if it is how to go about it.
I am not sure at all about (c)
Conclusion
Again many apologies for such a long post and many thanks to anyone who can help me with any of these problems.
