- #1

etotheipi

Two frames measure the position of a particle as a function of time: S in terms of x and t and S', moving at constant speed v, in terms of x' and t'. The acceleration as measured in frame S is $$ \frac{d^{2}x}{dt^{2}} $$ and that measured in frame S' is $$ \frac{d^{2}x'}{dt'^{2}} $$My question is how can we write the expression for the acceleration in frame S' in terms of that measured in frame S, noting the two coordinate transformations $$x' = \gamma(x-vt)$$ and $$t' = \gamma(t-vx)$$ I have had a go at the first derivative

\begin{align}

\frac{dx'}{dt'} &= \gamma \frac{dx}{dt'} - \gamma v \frac{dt}{dt'} \\ &= \gamma \frac{dx}{dt}\frac{dt}{dt'} - \gamma v \frac{dt}{dt'}

\end{align}

I tried deriving the time transformation with respect to t

\begin{align}

t' &= \gamma t - \gamma xv

\\ \frac{dt'}{dt} &= \gamma - \gamma v \frac{dx}{dt}

\end{align}

so that $$\frac{dt}{dt'} = \frac{1}{\gamma - \gamma v \frac{dx}{dt}}$$ but substituting this in doesn't really help and even then I'm not sure how to handle the second derivative. The aim would be to have some relationship like $$ \frac{d^{2}x}{dt^{2}} = f(\frac{d^{2}x'}{dt'^{2}})$$ where $$f$$ is some function. I was wondering if anyone could give me some tips or guidance since I don't really know if I'm going about this the right way. Thanks a bunch.

\begin{align}

\frac{dx'}{dt'} &= \gamma \frac{dx}{dt'} - \gamma v \frac{dt}{dt'} \\ &= \gamma \frac{dx}{dt}\frac{dt}{dt'} - \gamma v \frac{dt}{dt'}

\end{align}

I tried deriving the time transformation with respect to t

\begin{align}

t' &= \gamma t - \gamma xv

\\ \frac{dt'}{dt} &= \gamma - \gamma v \frac{dx}{dt}

\end{align}

so that $$\frac{dt}{dt'} = \frac{1}{\gamma - \gamma v \frac{dx}{dt}}$$ but substituting this in doesn't really help and even then I'm not sure how to handle the second derivative. The aim would be to have some relationship like $$ \frac{d^{2}x}{dt^{2}} = f(\frac{d^{2}x'}{dt'^{2}})$$ where $$f$$ is some function. I was wondering if anyone could give me some tips or guidance since I don't really know if I'm going about this the right way. Thanks a bunch.

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