The book states to stop particle B and calculate the velocity of A in B's frame.
i.e. transform velocity to a frame S' that means with velocity v(x) = UB = 0.7c

Transform UA to UA'

UA' = (UA - V) / [1 - (v/c^2)(UA)]

They then fill in values and obtain an answer of 0.23c. But I'm wondering where has this equation come from? I can see that it looks a bit like Lorrentz transformation gamma.

But if someone could explain exactly what's going on to obtain this equation? Has gamma been applied? Thank you

You know [itex]x' = \gamma (x-vt)[/itex] and [itex]t' = \gamma (t - \frac{v}{c^2}x)[/itex]

and you need to find out u' in terms of u

so its just simple maths ...

for the above 2 eqn's find dx' (by finding dx'/dt and taking dt to right side)and dt' (by finding dt'/dt and taking dt to right) and just divide them ...

So you differentiate x' with respect to t to obtain dx'/dt.
Then differentiate t' with respect to t to obtain dt'/dt.
Then you can just use chain rule as i mentioned above.
Remember to substitute u = dx/dt.

If you don't understand how to get it still, do say and i'll write it out if you want?
Also do you understand why this equation is used to solve the problem?

So u' would be the speed of the object viewed in frame S'. u would be the speed of the object view in frame S.

Now when considering these sorts of the problems, i like to consider Ship B to just be an inertial frame moving relative to another inertial frame (the Earth).

Let the Earth (stationary frame) to be frame S.

Now let Ship B be frame S', which is moving relative to S (Earth) at a constant velocity, v = 0.7c

Ship A is this object, that is moving at a velocity u = 0.8c in frame S (earth frame), you want to find it's velocity, u' in the frame S' (ship B frame)

So you just substitute these values in the equations to get your answer.

I meant equation not equations, sorry. The equation i was referring to is the same one you that i was helping you understand/derive from Lorentz transformations.

You can't use u' = u/gamma.

I think the relationship for length contraction is L' = L/gamma, you cannot use this to calculate relative velocities.

Also i accidently wrote v = 0 instead of v = 0.7c in my last post, have corrected that now.