About the mass-energy relation

[snoopies622]

I see how the premises

$$<br /> p = \gamma m v<br />$$

$$<br /> F = \frac {dp}{dt}<br />$$

and

$$<br /> <br /> W= \int F dx<br />$$

lead to

$$<br /> dW = mc^2 d \gamma<br />$$

and therefore

$$<br /> W = \gamma mc^2 + k<br />$$

where m is the rest mass and k is a constant of integration. But why do we conclude that k=0?

 

[Doc Al]

That constant won't equal 0. (The work done equals the KE, not the total energy.) Assume you start from rest and integrate to speed v.

 

[snoopies622]

I get

$$<br /> W = \gamma mc^2 - mc ^2 = ( \gamma - 1 ) mc^2 <br /> <br />$$

so

$$<br /> k = -mc^2 \neq 0<br />$$

Since W(v=0) = 0 this is indeed the kinetic energy and not the total energy. Thanks, Doc Al.

 

[snoopies622]

Follow-up: Am I to conclude that the constant of integration here represents the (negative) "rest energy" of the object, or is there a better way to arrive at that relationship?