# Relative velocities at high speed

• Coldie
In summary: To find the velocity of train A wrt train B, train B becomes your primed frame so choose your axes to make its velocity in the y 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.)
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.

Um... bump?

Coldie said:
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.)

## 1. What is relative velocity at high speed?

Relative velocity at high speed is the measurement of how fast an object is moving in relation to another object when both are traveling at high speeds.

## 2. How is relative velocity at high speed calculated?

To calculate relative velocity at high speed, the velocities of both objects must be added together using the formula v = u + w, where v is the relative velocity, u is the velocity of the first object, and w is the velocity of the second object.

## 3. How does relative velocity at high speed differ from relative velocity at low speed?

Relative velocity at high speed takes into account the effects of special relativity, which states that the speed of light is constant and the laws of physics are the same for all observers in uniform motion. At low speeds, these effects are negligible and relative velocity is calculated using simpler equations.

## 4. What are some real-world applications of relative velocity at high speed?

Relative velocity at high speed is important in fields such as astrophysics, where it is used to calculate the relative motion of celestial bodies. It is also used in aviation and space travel to determine the most efficient routes and trajectories for spacecraft and aircraft.

## 5. How does relative velocity at high speed affect time and distance measurements?

Due to the effects of special relativity, relative velocity at high speed can cause time and distance measurements to appear different for observers in different reference frames. This is known as time dilation and length contraction, and is important to consider in high-speed travel and communication.

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