- #1
Dakkers
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Homework Statement
A particle of mass m has a position along the x-axis as a function of time given by the equation
u = cgt / (1 + g2t2)1/2
where g is a constant and c is the speed of light.
(a) Find the 4-velocity of the particle.
(b) Express x and t as a function of the proper time of the particle.
(c) Find the 4-force acting on the particle. Does it ever exceed the speed of light?
Homework Equations
uμ = γ(c, u)
λ = dt / dτ
The Attempt at a Solution
a) Given the first equation, the four-velocity is simply
uμ = γ(c, cgt / (1 + g2t2)1/2
I think.
b) To find the position, we take the integral of dx/dt and find
x = (c/g)(1 + g2t2)1/2
If we let dt = γdτ, then we can easily see t = γτ.
However, this is a problem, as the particle is accelerating (its second time derivative is not zero) and that means γ must change. And, another problem, is that we cannot sub in
γ = (1 - u2/c2)-1/2
because then we have a recursive definition.
c) I know that the four-force is simply mass x second derivative of the four position (or mass x derivative of four-velocity), but I am not too sure how to differentiate it. I also know that the four-force is (F0, F) but I don't know how to find F0 :(
PLEASE BE GENTLE. I am a first-year physics student, and my university decided to put general relativity into a first-year course.