Transformation of acceleration between two reference frames

1. May 15, 2014

Paul Black

hello

i want to derive the Transformation of acceleration between two reference frames
i searched in internet and found a book but i dont understand just one step

i attach a picture so you can see what i found in the internet

my problem is eq. (1.10) \begin{align}du=\frac{dU}{\gamma ^{2}(1-U\frac{v}{c^{2}})^{2}}\end{align}
how did he get this factor
can you please give me a tip

thank you very much

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2. May 15, 2014

Bill_K

There's a misprint. Calculus rule is d(x/y) = (y dx - x dy)/y2 where x = U - v and y = 1 - Uv/c2, so the first denominator in Eq.(1.10) should be (1 - U v/c2)2. The numerator is dU(1 - v2/c2), which reduces to dU(1/γ2).

3. May 15, 2014

Paul Black

where is this misprint? is it the first denominator? the square is missing right?
my problem is this gamma in eq 1.10 how did he get it? i dont understand it
also the next step with gamma
can you show me how to get it?
thank you

4. May 15, 2014

xox

There are TWO errors, not one in the book:

$$dU=\frac{dU(1-Uv/c^2)-(U-v)(-dUv/c^2)}{(1-Uv/c^2)^2}$$

The final result is correct, the derivation is riddled with errors.

Last edited: May 15, 2014
5. May 15, 2014

Paul Black

xox thank you for your answer but im sorry you are wrong
the positive sign is wrong. it must be negative
therefore the only one error is the square in the denominator

but this is not my problem. i just want to know how to get the next step with gamma square.
hope you can help me

6. May 15, 2014

xox

There are TWO minus signs, they indeed cancel each other, so I was a little pedantic.

$$du=\frac{dU(1-Uv/c^2)-(U-v)(-dUv/c^2)}{(1-Uv/c^2)^2}=\frac{dU(1-v^2/c^2)}{(1-Uv/c^2)^2}$$

But $$1-v^2/c^2=\frac{1}{\gamma^2}$$

Last edited: May 15, 2014
7. May 15, 2014

Paul Black

yes you are right
sorryyyy

this gamma confused me xD

8. May 15, 2014

xox

You are welcome!

9. May 15, 2014

Paul Black

xox
thank you very much. i get it
so now i have to divide both sides by dt to get du/dt=a
but again with this gamma. eq 1.11 i dont understand it

10. May 15, 2014

xox

OK, let's help u a little more:

We agree that:
$$du=\frac{dU}{\gamma^2(1-Uv/c^2)^2}$$

Now,

$$dt=\gamma(dT-dXv/c^2)$$

Therefore:

$$a=\frac{du}{dt}=\frac{dU}{\gamma^3(1-Uv/c^2)^2(dT-dXv/c^2)}=\frac{dU/dT}{\gamma^3(1-Uv/c^2)^2(1-dX/dTv/c^2)}=\frac{dU/dT}{\gamma^3(1-Uv/c^2)^3}=\frac{A}{\gamma^3(1-Uv/c^2)^3}$$

If $u=V$ then $a=\gamma^3A$

11. May 15, 2014

Paul Black

thank you i got it finally
thank you very much

12. May 15, 2014