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* Could you help to integrate
dE = (m+dm)*v*dv + (c^2)*dm
I would like to compare the result with relativistic kinetic energy
which as you know is calculated by equation
E = mc^2( 1/(1-v^2/c^2)^(1/2) -1 )
Kinetic energy - Wikipedia, the free encyclopedia
Thank you very much for any possible help.
I am sorry, likely my initial question was wrong formulated.
Maybe this question would be more correct and easier:
How to prove
dm = m*v*dv / (c^2 - v^2)
is equivalent to
m = m0 / (1-(v^2 / c^2))^(1/2)
?
It is easy to write a program to calculate
dm = m*v*dv / (c^2 - v^2)
numerically by expression
m += m*v*dv / (c^2 - v^2)
inside a loop.
But I need analytical derivation.
Thank you
It's a fairly straight-forward differential equation. I'm not sure I agree with your results. Here's WolframAlpha's solution (this DE is a first-order separable ODE, with decently straight-forward integrals).
Thank you very much. It looks it works well.Quote:
Originally Posted by Ackbeet
Now I can write better equation for kinetic energy:
dE = ( m*c^2*v / (c^2 – v^2) ) dv = ( (m0/(1-v^2/c^2))*c^2*v / (c^2 – v^2) ) dv
I think this is equal to relativistic kinetic energy
E = m0*c^2( 1/(1-v^2/c^2)^(1/2) -1 )
I was mentioned by my first post.
By using WolframAlpha I have got :
E = m0*c^2 / (1-v^2 / c^2)^(1/2) + constant.
http://www.wolframalpha.com/input/?i=int+m_0*c^2*v%2F%28%281-v^2%2Fc^2%29^%281%2F2%29*%28c^2+-+v^2%29%29+dv
Maybe you have some hint how to prove this constant equals to
-m0*c^2 ?
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Thank you, I have solved it by myself. You may close the tread.
Glad you got your solution.
We don't generally close threads around here unless there's something wrong with them. I have marked your thread as "SOLVED", though.
Cheers.