Could you help to integrate dE = (m+dm)*v*dv + c^2*dm

* 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.

Re: Could you help to integrate dE = (m+dm)*v*dv + c^2*dm

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

Re: Could you help to integrate dE = (m+dm)*v*dv + c^2*dm

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).

Re: Could you help to integrate dE = (m+dm)*v*dv + c^2*dm

Quote:

Originally Posted by

**Ackbeet** 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.

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 ?

----------------------------------------------------------------------------

Thank you, I have solved it by myself. You may close the tread.

Re: Could you help to integrate dE = (m+dm)*v*dv + c^2*dm

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.