Relative velocities at high speed

[Coldie]

Hi again,

Our textbook gives us equations to find the speed of objects in relation to others in the x, y, and z planes. These are:

$$v^'_x = \frac{v_x - u}{1 - uv_x/c^2}$$
$$v^'_y = \frac{v_y}{\gamma(1 - uv_x/c^2)}$$
$$v^'_z = \frac{v_z}{\gamma(1 - uv_x/c^2)}$$

My first question is why is the equation for velocity in the x direction different from those in the y and z directions? Since all directions are relative, that would mean that simply turning 90º in any direction would mean that you'd have to use a different equation to find the velocity of something in a given axis?

My next question is from one of the homework problems, which says:

A and B are trains on perpendicular tracks. The velocities are in the station frame (S frame).
a) Find $$v_AB$$, the velocity of train B with respect to train A.
b) Find $$v_BA$$, the velocity of train A with respect to train B.

The picture shows that train A is going directly upwards from the train station at .8c, and train B is moving directly to the right from the station, also at .8c.

Now, just looking at the equations to finding the answer to part a you can tell that something's not right. The equation to find $$v^'_y$$ has [v_y] on the top, which in this case is 0 since train B is not moving vertically, so following the given equations, the vertical speed of train B relative to A is 0, which is not correct. The answer in the back is arrived at by using the equation for $$v^'_x$$ to find $$v^'_y$$ and using the equation for $$v^'_y$$ to find $$v^'_x$$. Can someone tell me how you're supposed to know when to switch the x and y axes to use the proper equations to find the answer?

I hope I've made everything clear, thanks for the help.

 

[Coldie]

Um... bump?

 

[Doc Al]

My first question is why is the equation for velocity in the x direction different from those in the y and z directions? Since all directions are relative, that would mean that simply turning 90º in any direction would mean that you'd have to use a different equation to find the velocity of something in a given axis?

But all directions are not equal!** These equations assume that the frame is moving with speed u along the +x axis. So, you must redefine your axes accordingly to make use of these formulas.

To find the velocity of train B wrt train A, train A becomes your primed frame so choose your axes to make its velocity in the x direction.

(**Recall that length contracts along the direction of motion but not perpendicular to it--thus the direction of motion selects one axis for special treatment.)