Relative velocity in special relativity

In summary, the conversation discusses the relative velocity of two aeroplanes observed from the ground, and how to express this velocity in terms of their individual velocities. Using the general Lorentz transformation and the equation for the square of the vector, the final solution is derived as an expression depending only on the velocities of the two planes. Further simplification can be done by expressing gamma in terms of the velocity u.
  • #1
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Homework Statement


Imagine we are observing two aeroplaes from the ground and let their velocities be ##\mathbf{u}## and ##\mathbf{v}## respectively. Assume that the first plane has radar equipment permitting a measurement of the speed of the other plane relative to itself. The velocity so measured will be the relative velocity of our definition. We must express this relative velocity in terms of the components of the velocities ##\mathbf{u}## and ##\mathbf{v}## of the two planes, as observed from the ground. The velocity of the second plane measured from the ground is $$\mathbf{v}=\frac{d\mathbf{r}}{dt}$$ while its velocity measured from the other plane is $$\mathbf{v}^{\prime}=\frac{d\mathbf{r}^{\prime}}{dt^{\prime}}.$$Using the general Lorentz transformation we have $$\mathbf{v}^{\prime}=\frac{\mathbf{v}-\mathbf{u}+\left(\gamma-1\right)\left(\frac{\mathbf{u}}{u^{2}}\right)\left\{ \left(\mathbf{u}\cdot\mathbf{v}\right)-u^{2}\right\} }{\gamma\left(1-\mathbf{u}\cdot\mathbf{v}\right)}$$where$$\gamma=\frac{1}{\sqrt{1-u^{2}}}.$$Calculate the square of the vector ##\mathbf{v}^{\prime}##.

Homework Equations


$$\left(\mathbf{u}\times\mathbf{v}\right)^{2}=u^{2}v^{2}-\left(\mathbf{u}\cdot\mathbf{v}\right)^{2}$$

The Attempt at a Solution


The solutions is $$v^{\prime2}=1-\frac{\left(1-u^{2}\right)\left(1-v^{2}\right)}{\left(1-\mathbf{u}\cdot\mathbf{v}\right)^{2}}=\frac{\left(\mathbf{u}-\mathbf{v}\right)^{2}-\left(\mathbf{u}\times\mathbf{v}\right)^{2}}{\left(1-\mathbf{u}\mathbf{\cdot}\mathbf{v}\right)^{2}}.$$Taking the square of the vector ##\mathbf{v}^{\prime}## I have $$v^{\prime2}=\frac{1}{\gamma^{2}\left(1-\mathbf{u}\cdot\mathbf{v}\right)^{2}}\left\{ \left(\mathbf{v}-\mathbf{u}\right)^{2}+\frac{1}{u^{2}}\left(\gamma-1\right)^{2}\left(\mathbf{u}\cdot\mathbf{v}-u^{2}\right)^{2}+\frac{2}{u^{2}}\mathbf{u}\cdot\mathbf{v}\left(\gamma-1\right)\left(\mathbf{u}\cdot\mathbf{v}-u^{2}\right)-2\left(\gamma-1\right)\left(\mathbf{u}\cdot\mathbf{v}-u^{2}\right)\right\}.$$ Can someone help me?
 
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  • #2
It is an answer that depends on v and u only, that is good. Now you can simplify it (don't forget to express gamma via u). Not nice, and there could be a shorter way, but it should work.
 
  • #3
It works! Thanks
 

Related to Relative velocity in special relativity

1. What is relative velocity in special relativity?

Relative velocity in special relativity refers to the measurement of the speed of an object relative to another object, taking into account the effects of time dilation and length contraction as predicted by Einstein's theory of relativity.

2. How is relative velocity calculated in special relativity?

In special relativity, relative velocity is calculated using the Lorentz transformation equations, which take into account the relative motion between two objects and the speed of light. These equations allow for the calculation of relative velocities that are different from those predicted by classical Newtonian physics.

3. Does relative velocity have an upper limit in special relativity?

Yes, according to special relativity, the speed of light is the ultimate speed limit in the universe. This means that no object can have a relative velocity greater than the speed of light. This is a fundamental principle of the theory and has been confirmed by numerous experiments and observations.

4. How does relative velocity in special relativity affect the perception of time?

One of the consequences of relative velocity in special relativity is time dilation, which means that time moves slower for objects that are moving at high speeds relative to an observer. This effect has been observed in experiments with high-speed particles and is a key aspect of the theory of relativity.

5. Can relative velocity in special relativity be observed in everyday life?

While the effects of relative velocity and time dilation are not typically noticeable in everyday life, they can be observed in certain situations, such as in GPS satellites and high-speed particle accelerators. However, in our day-to-day experiences, the effects are so small that they are imperceptible to us.

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