Uniformly Accelerated Motion
المؤلف:
Sidney B. Cahn And Boris E. Nadgorny
المصدر:
A GUIDE TO PHYSICS PROBLEMS
الجزء والصفحة:
part 1 , p 38
13-7-2016
1593
Uniformly Accelerated Motion
Determine the relativistic uniformly accelerated motion (i.e., the rectilinear motion) for which the acceleration ω0 in the proper reference frame (at each instant of time) remains constant.
a) Show that the 4-velocity
b) Show that the condition for such a motion is
where ω0 is the usual three dimensional acceleration .
c) Show that in a fixed frame (b) reduces to
d) Show that
Do these expressions have the correct classical behavior as c → ∞.
SOLUTION
a) The 4-velocity, by definition
where 
Therefore,
or
b) For arbitrary velocity v, the 4-acceleration
For μ = 0
For μ = 1, 2, 3
Therefore, we find
(1)
In the proper frame of reference, where the velocity of the particle v = 0 at any given moment, and assuming
Using ω2 = ωμωμ, we have
c) ωμ ωμ from (1) may be written in the form
(2)
Using the identity (A × B)2 = A2 B2 – (A.B)2, we may rewrite (2)
(3)
In the fixed frame, since the acceleration is parallel to the velocity, (3) reduces to
So, given that ω2 is a relativistic invariant, 
Now, differentiating
(4)
d) Integrating (4), we have
(5)
Taking v = 0 at t = 0, we obtain
and so
(6)
As ω0t → ∞, v → c. Integrating (6) with yields x(0) = 0
(7)
As c → ∞ (classical limit), (6) and (7) become
appropriate behavior for a uniformly accelerated classical particle.
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