Relativistic rocket equation question thingy

[Sejanus]

I found this question challenging and enjoyable;



It's from a scholarship calculus exam I sat.
Post your solutions/attempts! Mine was rather long winded, I hope someone finds a more elegant method.
I have more pictures zoomed in if you can't see these properly.

[pocaracas]

This looks challenging... maybe I'll look into it when I get home, some 6 hours from now...

[Sejanus]

I thought this was going to be one of those threads that just dies.

[Anomalocaris]

What am I solving for?

[Sejanus]

Basically you're solving for (delta v).

[pocaracas]

Deceptively simple...


If there's anything you can't understand, I can take a better photo...

edit: bear in mind that ln = log_e

[FreeTony]

You can do it in a few lines if you solve for delta v, then just use the identity a^b = e^(ln(a^b)) = e^(b*ln(a))

b = 2u/c and a = m0/m1

All these types of problems can generally be solved with the identities ln(a*b) = ln(a) + ln(b) and ln(a^b) = b*ln(a)

[Sejanus]

Deceptively simple...

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If there's anything you can't understand, I can take a better photo...

edit: bear in mind that ln = log_e

Nice job! you did it slightly different than I did, but we still used roughly the same process.

You can do it in a few lines if you solve for delta v, then just use the identity a^b = e^(ln(a^b)) = e^(b*ln(a))

b = 2u/c and a = m0/m1

All these types of problems can generally be solved with the identities ln(a*b) = ln(a) + ln(b) and ln(a^b) = b*ln(a)

What you're describing here seems to be roughly the same as what I did, and it takes more than "a few lines". If you mean something other than I understand, please post a picture of your solution.

[Sejanus]

Here's my solution. I skipped a few lines here and there, I can explain those if its not obvious.

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Also, here is the second part of the question!

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The relativistic rocket equation is derived from the following differential equation, where u and c are constants.
Show that:
is a solution to this differential equation.

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