Hi, I'm working out of Hartle's book Gravity and had a question about Ch. 5 problem 7. The problem states that a particle has constant acceleration along the x-axis, that is, the acceleration when measured in it's instantaneous rest frame is always the same constant. They want me to get the parametric equations for x and t as functions of proper time [itex]\tau[/itex], assuming at t=0 that x=0 and v=0.(adsbygoogle = window.adsbygoogle || []).push({});

I assumed that since the acceleration is constant, the three-force is also constant. I can then use

[tex]

\frac{dp}{dt} = F = ma

[/tex]

for some constant a. Since F is constant I can integrate once to get

[tex]

{\gamma}m\frac{dx}{dt} = mat

[/tex]

Or, solving for [itex]\frac{dx}{dt}[/itex]

[tex]

\frac{dx}{dt} = \frac{at}{\sqrt{1+a^2t^2}}

[/tex]

Using the relationship [itex]{d\tau}^2 = dt^2 - dx^2[/itex] I can arrive at

[tex]

d\tau = dt\sqrt{(\frac{1}{1+a^2t^2})}

[/tex]

Integrating and solving for t and letting [itex]\tau[/itex] = 0 when t=0 I get

[tex]

t(\tau) = \frac{{\sin}h(a\tau)}{a}

[/tex]

I can then solve for x and get

[tex]

x(\tau) = \frac{{\cos}h(a\tau)}{a} - \frac{1}{a}

[/tex]

due to the initial conditions. Did I do this right?

EDIT: I really just need to know if my first couple steps are right, as everything follows from that.

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# Homework Help: Relativity - Constant Acceleration

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