3.3 Accelerated Motion
25
3.3 Accelerated Motion
Now we are ready to study the trajectory of an accelerated particle in one space
dimension. We will think of the particle as a small rocket, since a rocket is built with
internal means of acceleration. In doing this we will see how convenient the concept
of rapidity is for such calculations (Misner 1973).
Consider a rocket moving in the x direction as in Fig. 3.3. The proper time τ
provides a convenient parameter for defining the trajectory of the rocket, ct(τ ), x(τ ).
At proper time τ the rocket has velocity v in the lab system S, while in its instantaneous rest frame S
its velocity is of course zero. A short time dτ later its velocity in
S
is given in terms of the acceleration by
dv
= adτ, dβ
= (a/c)dτ, a = proper acceleration.
(3.21)
The proper acceleration is that measured in the proper frame, where the rocket is
instantaneously at rest. In the lab frame S the rocket velocity after the little time
interval and velocity change is gotten from the velocity addition relation in Exercise
1.3,
β(after dτ ) =
β + dβ
/
1 + βdβ
.
(3.22)
Thus the change in the lab velocity of the rocket to first order in dβ
is
dβ =
1 − β
2
dβ
= dβ
/γ
2
= (a/c)dτ/γ
2
.
(3.23)
This is a differential relation giving β as a function of τ and the proper acceleration
since γ is a function of β; if the acceleration were given as a function of τ we could
integrate (3.23) to get β(τ ).
However there is a more elegant way to analyze (3.23) in terms of rapidity. From
the definition of rapidity θ in (1.28) we have
β = tanh θ, dβ = sech
2
θ dθ = dθ/cosh
2
θ = dθ/γ
2
,
dθ = γ
2 dβ.
(3.24)
Fig. 3.3 Trajectory of the accelerated particle or rocket