Lorrentz velocity transformation

[Mangoes]

I'm being asked to derive the velocity transformation between v_y and v_y' and my result isn't exactly matching my goal but I don't know what I'm doing wrong. It's an introductory modern physics course and we're covering special relativity.

Assume a reference frame S' moving in some constant velocity v_x with respect to a stationary frame S. Since time isn't synchronized (dt ≠ dt') then the velocities v_y and v_y' aren't the same even though y and y' aren't being affected by length contraction (y = y'). The aforementioned statement isn't necessary for the following but I'm just wondering if the above is a correct statement.

Anyways, this is my logic:
Since y = y', dy = dy'. I want to know dy/dt'. From the chain rule:

$$\frac{dy}{dt'} = \frac{dy}{dt}\frac{dt}{dt'}$$

Letting $$λ = \frac{1}{(1 - (v/c)^2)^{1/2}}$$, the Lorrentz transformation for time is:

$$t' = λ(t - \frac{vx}{c^2})$$

Since there is no fundamental difference between two inertial reference frames, the inverse transformation must be of the same form:

$$t = λ(t' + \frac{vx'}{c^2})$$

In differential form,

$$dt = λ(dt' + \frac{vdx'}{c^2})$$

Now differentiating the above wrt t',

$$\frac{dt}{dt'} = λ(\frac{dt'}{dt'} + v\frac{dx'}{dt'}/c^2)$$

$$\frac{dt}{dt'} = λ(1 + vv_x'/c^2)$$

Going back to my original goal:

$$\frac{dy}{dt'} = \frac{dy}{dt}\frac{dt}{dt'} = v_y\frac{dt}{dt'}$$

$$\frac{dy}{dt'} = v_y(λ(1 + vv_x'/c^2))$$

...which is wrong. Where am I going wrong with this? Would appreciate some clarity.

 

[Orodruin]

Does the object have a velocity component also in the x-direction in S? What is $v_x'$ if $v_x = v$?

 

[Mangoes]

If I'm understanding correctly, yes, the object would have a velocity component in S. I just realized I should have been a little more clear in my first post.

The origin O' in S' is moving along with some velocity v_x and an observer at rest in S' is observing an object P in S' to be moving in a direction parallel to the direction of S' with velocity u.

An observer in a stationary frame S would then see P's velocity to be given by the Lorrentz velocity transformation,

$$v_x' = \frac{v_x - u}{1 - \frac{uv_x}{c^2}}$$

The inverse transform for v_x in terms of v_x would be obtained by swapping primes and replacing u by -u.

The more I think about it, the more I get confused though. If P is moving only in the direction parallel to motion of S' and both observers in S and S' agree that y = y', I'm not seeing how it's possible for P to be observed to have a velocity perpendicular to v_x in either frame S or S'. I think I'll just have to ask for clarification in lecture because I think I must have misheard or misunderstood something in the premise.

The result I'm being asked should be

$$v_y' = \frac{v_y(1 - (v/c)^2)^{1/2}}{1 - \frac{uv_x}{c^2}}$$

 

[Doc Al]

The origin O' in S' is moving along with some velocity v_x and an observer at rest in S' is observing an object P in S' to be moving in a direction parallel to the direction of S' with velocity u.

I think you have those mixed up. The origin O' is seen to move with velocity u in frame S (parallel to the x axis). The object moves with velocity v'_x in S' and velocity v_x in S.

I'm not seeing how it's possible for P to be observed to have a velocity perpendicular to v_x in either frame S or S'.

It's not. You're mixing yourself up. If P is moving only in the x' direction according to S', then it will be moving only in the x direction according to S.

 

[Mangoes]

I think you have those mixed up. The origin O' is seen to move with velocity u in frame S (parallel to the x axis). The object moves with velocity v'_x in S' and velocity v_x in S.

Yes, you're right. Thank you. I switched around v_x and u.

It's not. You're mixing yourself up. If P is moving only in the x' direction according to S', then it will be moving only in the x direction according to S.

Yeah, I would think not. I'll just ask when I get the chance to clarify the premise. I must have misunderstood or misheard something.