Are Lorentz transformations consistent with different reference frames?

[toph]

Question
Show that, with V = 4/5c, the Lorentz transformation of the equations, t^prime = y(V) (t-(v/c^2)x) and x^prime = y(V) (x-Vt). (where y(V) = the Lorentz factor).

can be written as

ct^prime = 5/3ct - 4/3x

and

x^prime = 5/3x - 4/3ct
Relevant equations
y(V) = 1/(sqrt1-(V/c)^2)
The attempt at a solution
I have calculated y(V) = 5/3 (if V = 4/5c)
and i can see how the left hand term in each equation becomes 5/3ct and 5/3x respectivley. But i can't figure where the 4/3 term comes from?? or how to derive it?

 

[neutrino]

The second term on the RHS in both equations contain a 'v', right?

 

[George Jones]

$$\left( \frac{5}{3} \right) \left( \frac{4}{5} \right) = \frac{4}{3}.$$

 

[toph]

Ahh... the penny drops. thank you

 

[toph]

A further Lorentz transformation problem.

The question i have is.

Use these Lorentz transformations ct'=5/3ct-4/3x and x'=5/3x-4/3ct. to determine the (ct', x') coordinates, in meters, that observer O' assigns to events e1 and e2.

relevant equations and information.

from a previous question i have determined the coordinates of the events in the rest frame of observer O to be e1 = (0, 240)m, e2 = (60, 240)m.

My attempt at answer

Using the given Lorentz transformations i have found for event e1 as observed by O' is.

e1 = (ct', x') = (5/3ct-4/3x, 5/3x-4/3ct) = (-180, 144)m
e2 = (ct', x') = (5/3ct-4/3x, 5/3x-4/3ct) = (-144, 99) m

However i feel uncomfortable with these answers, but cannot put my finger on why? Please can some one check my results?

thank you

 

[toph]

Further to my last post, the reason i feel uncomfortable is that surely the x component of the coordinates should be the same for both events? I can check this via the Lorentz length contraction formula, which gives l = lo/y(V) =144?

 

[neutrino]

It seems to be errors in arithmetic to me.

Further to my last post, the reason i feel uncomfortable is that surely the x component of the coordinates should be the same for both events? I can check this via the Lorentz length contraction formula, which gives l = lo/y(V) =144?

Surely, then, both observers must be referring to the same reference frame.

 

[toph]

It seems to be errors in arithmetic to me.
Surely, then, both observers must be referring to the same reference frame.

yep your right i have just spotted my error. i think i have also calculated the transformed event coordinations incorrectly as well?

 

[neutrino]

Surely, then, both observers must be referring to the same reference frame.

Sorry about my previous post. I misread your statement, and gave a stupid reply (eyes can play tricks on you late at night!). I really cannot say whether the x coordinates of the two events in the O frame are the same or not, unless I have more information. But if you have managed to find you errors in spite of my comment, then well and good.