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Homework Statement
Show that the energy-momentum relationship, E^2 = p^2 * c^2 + (m*c^2)^2, follows from the expressions E = (gamma)*m*c and p = (gamma)*m*u
where
(gamma) = 1 / sqrt(1 - (u^2)/(c^2)) the lorentz transformation factor.
m is the rest mass.
c is the speed of light
u is the velocity of the particle
E is the total energy
p is the momentum
The book does not teach about relativistic mass, so I think I supposed to derive this without making a distinction between m and m0.
Homework Equations
(1): E^2 = p^2 * c^2 + (m*c^2)^2
(2): E = (gamma)*m*c
(3): p = (gamma)*m*u
The Attempt at a Solution
When the chapter introduces the formula E^2 = p^2 * c^2 + (m*c)^2, it does not show how it derived this equation. Instead it says that it just says "By squaring [equations (2) and (3)] and subtracting, we can eliminate u. The result after some algebra is [equation (1)]."
My first attempt was to start by squaring both sides of equation (3).
p^2 = (gamma)^2*m^2*u^2
Then get it in terms of m^2.
m^2 = (p^2)/(u^2) - (p^2)/(c^2)
In equation (2), square both sides, then substitute m^2 to get:
E^2 = (gamma)^2 * [(p^2)/(u^2) - (p^2)/(c^2)] * c^2
After some algebra I got:
E^2 = [(gamma)^4 * m^2 * c^4] - [(gamma)^2 * p^2 * c^2]
This is nearly what I'm trying to derive, however, the (gamma) terms are still there and I don't know how to get rid of them.
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