"Astronomical Calculus" Spaceship Dilation problem

[Sabertooth]

Homework Statement

Astronomical Calculus:Spaceship Dilation problem

Relevant Equations

t'=t(√-1(v^2/c^2)). Integral Calculus.

Hi everyone. I have provided myself a problem that I insist on solving, however, I want to do it "the right way" where I can put every parameter into a calculator and get an answer quickly. I pondered doing it manually and figured that it could be done to a reasonable precision in an hour or two, but that is not enough for me to consider this problem solved. I want to use formal calculus to put the parameters into an equation and get the answer.

Is there any calculus aficionados that could help me get all the ingredients for solving this problem here, and getting a result. This problem relates purely to Special Relativity & Velocity⋅. General Relativity can be ignored for all intents and purposes.

 

[PeroK]

As you don't have explicit formulas for the speed profiles, I suggest this is a numerical exercise.

By the way, what you are asked to calculate here is differential ageing and not, strictly speaking, time dilation.

 

[Sabertooth]

As you don't have explicit formulas for the speed profiles, I suggest this is a numerical exercise.

By the way, what you are asked to calculate here is differential ageing and not, strictly speaking, time dilation.

No formulas for the speed profile, but all the information can be read directly off the graph. Can we plug a picture of the graph into a program and retrieve the formula?

 

[PeroK]

No formulas for the speed profile, but all the information can be read directly off the graph. Can we plug a picture of the graph into a program and retrieve the formula?

I doubt it, but you could always search the web for such a program.

 

[Sabertooth]

I doubt it, but you could always search the web for such a program.

I'll try and look for one. Why do you doubt its existence? It seems like something a computer should be able to do no problem.
Also, if we had this formula for the speed, what would the calculus look like formally?

 

[PeroK]

I'll try and look for one. Why do you doubt its existence? It seems like something a computer should be able to do no problem.
Also, if we had this formula for the speed, what would the calculus look like formally?

You could look for a program to do that too!

 

[Sabertooth]

hmm, not a lot of help from the certified science advisor it seems :/
It's okay if you're not in a helpful mood. There is always tomorrow.

 

[PeroK]

hmm, not a lot of help from the certified science advisor it seems :/
It's okay if you're not in a helpful mood. There is always tomorrow.

You have to show some effort to solve the problem yourself.

 

[Sabertooth]

You have to show some effort to solve the problem yourself.

Which I'm definitely prepared to do, but from what I gathered from your reply I need some type of formula that isn't present. Can the formula or the function (the thing that is missing) be calculated manually with some type of integral or derivative?

 

[PeroK]

Which I'm definitely prepared to do, but from what I gathered from your reply I need some type of formula that isn't present. Can the formula or the function (the thing that is missing) be calculated manually with some type of integral or derivative?

What do you know about time dilation/differential ageing?

 

[Sabertooth]

What do you know about time dilation/differential ageing?

Pretty much everything there is to know on that front. It's the calculus part bothering me, as I need to be able to do the dilation calculations when velocities are varying with time in a smooth curve.
I would also be able to apply this answer to dozens of different scenarios and problems that I've been thinking about, so that's why I've set up the problem in this fashion.

 

[PeroK]

Pretty much everything there is to know on that front. It's the calculus part bothering me, as I need to be able to do the dilation calculations when velocities are varying with time in a smooth curve.
I would also be able to apply this answer to dozens of different scenarios and problems that I've been thinking about, so that's why I've set up the problem in this fashion.

As you know we have $d\tau(t) = \frac {dt} {\gamma(t)}$. You integrate that.

 

[Sabertooth]

Is that the notation we use to say that we need to divide the curve up into n number of sections and calculate the lorentz factor for each section, then add them together and compare? Let's say we split it up into 10 000 sections. That should give us reasonable accuracy, I presume. Because the acceleration is rather smooth in this graph.

 

[PeroK]

Is that the notation we use to say that we need to divide the curve up into n number of sections and calculate the lorentz factor for each section, then add them together and compare? Let's say we split it up into 10 000 sections. That should give us reasonable accuracy, I presume. Because the acceleration is rather smooth in this graph.

Yes, that's a numerical approach using the data from the graph.

 

[Sabertooth]

since it would take way too many hours to calculate 10 000 sections manually what approach can we use to make it work?

 

[PeroK]

since it would take way too many hours to calculate 10 000 sections manually what approach can we use to make it work?

I suspect you are supposed to use the squares on the graph to extract the data for 30-40 points. You need a spreadsheet or a program.

 

[etotheipi]

As a first approximation, you could curve fit a quartic to 5 points on the first graph and a quadratic to 3 points on the second graph to find $v(t)$.

Then fire up wolfram alpha and see if numerically integrates $\frac{dt}{\gamma}$