# Using rapidity - prove velocity transformation equations

• zimo
In summary, the law of transformation of velocities can be proved using the rapidity equations and the identity for tanh(a+b). The equations for Vx, Vy, and Vz can be derived by considering the relation between distances in different coordinate systems.
zimo

## Homework Statement

Prove the law of transformation of velocities

$$\begin{array}{l} {v_x} = \frac{{{v_x}^\prime + V}}{{1 + {v_x}^\prime V/{c^2}}}\\ {v_y} = \frac{{{v_y}^\prime }}{{\gamma (1 + {v_x}^\prime V/{c^2})}}\\ {v_z} = \frac{{{v_z}^\prime }}{{\gamma (1 + {v_x}^\prime V/{c^2})}} \end{array}$$

## Homework Equations

Rapidity equations

## The Attempt at a Solution

I proved Vx with an ease using rapidity equations and the identity for tanh(a+b)
now, I'm stuck with revealing the relation to Vy and Vz.

What happens if you use the same method you used to prove the first equation, except taking into account that a distance in some direction orthogonal to V is the same in both coordinate systems?

## 1. What is rapidity and how does it relate to velocity transformation equations?

Rapidity is a measure of the rate at which an object is moving in relation to the speed of light. It is often used in special relativity to calculate the velocity of an object as it approaches the speed of light. Velocity transformation equations use rapidity to describe how the velocity of an object changes as seen from different reference frames.

## 2. How do we prove velocity transformation equations using rapidity?

To prove velocity transformation equations using rapidity, we start with the Lorentz transformation equations, which describe the relationship between the coordinates of an event as seen from different reference frames. By substituting the rapidity variable for the velocity variable, we can derive the velocity transformation equations.

## 3. What is the significance of using rapidity in velocity transformation equations?

Using rapidity in velocity transformation equations allows us to accurately describe the behavior of objects moving at high speeds, including the effects of time dilation and length contraction predicted by special relativity. It also simplifies the equations by removing the need for the Lorentz factor and instead using a single variable.

## 4. Can the velocity transformation equations be used for objects moving at any speed?

No, the velocity transformation equations are only valid for objects moving at relativistic speeds, meaning speeds close to the speed of light. At lower speeds, the equations can still be used, but the effects of special relativity become negligible and the classical velocity addition formula is a more accurate representation.

## 5. Are there any limitations to using rapidity in velocity transformation equations?

One limitation of using rapidity in velocity transformation equations is that it assumes objects are moving in a straight line at a constant velocity. It cannot accurately describe the behavior of objects accelerating or changing direction. Additionally, it only applies to objects moving in a vacuum and does not take into account external forces or interactions with other objects.

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