Compute relativistic addition of velocities

In summary, the conversation discusses the coordinates of particles in a spacetime diagram and their corresponding events in different frames of reference. The goal is to compute the ratio of distance to time, which is found to be (u+v)/(1+uv). The conversation also covers a step-by-step calculation to arrive at this result.
  • #1
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Homework Statement
A particle moves to the left with speed u from the laboratory's frame. Another particle moves to the right with speed v. Both particles are related at the same time from the same place in the laboratory. From the perspective of the first particle, what is the speed of the second particle?
Relevant Equations
Spacetime lengths are conserved ##\Delta s^2=\Delta t^2-\Delta x^2##
edit: I had a sign error that is now corrected, no further help needed.

Consider in the laboratory frame one second passing. Let ##(t,x)## be the coordinates of various events in a spacetime diagram. Both particles are released at ##(0,0)## Then after one second the left moving particle is at ##(1,-u)##. The right moving particle is at ##(1,v)##.

Now consider the spacetime diagram the first particle draws. Both particles are released at ##(0,0)## still. The event of the particle reaching ##(1,-u)## in the laboratory frame corresponds to reaching ##(t_0,0)## in the particle frame, where ##t_0^2-0^2=1-u^2##.

The event of the rightward particle reaching ##(1,v)## in the laboratory frame corresponds to reaching a point ##(t,x)## in the particle frame. Our goal is to compute ##x/t## which I believe should be ##\frac{u+v}{1+uv}##. We know the spacetime distance between ##(t,x)## and the origin is preserved, and this gives us ##t^2-x^2=1-v^2##. We also know the spacetime interval to the end of the other particle's path which gives us ##(t-t_0)^2-x^2= -(u+v)^2##.

Edit: I wrote this whole thing up with a ##+(u+v)^2## which was the source of all my problems.To try to solve for ##t## and ##x##, I tried subtracting the second equation from the second to get
$$(t-t_0)^2-t^2=-(u+v)^2-(1-v^2)$$
$$-2t_0t+t_0^2=-(u+v)^2-(1-v^2)$$
$$t=\frac{(u+v)^2+1-v^2+t_0^2}{2t_0}$$
We can use ##t_0^2=1-u^2## in the numerator and expand the ##(u+v)^2## to get
$$t=\frac{1+uv}{\sqrt{1-u^2}}$$

Ok, cool. Now we also have ##x2=t^2-1+v^2## from which we get

$$x^2/t^2=1-\frac{(1-v^2)(1-u^2)}{(1+uv)^2}$$
Putting the 1 over a common denominator and expanding the numerator gives
$$x^2/t^2= \frac{1+2uv+u^2v^2-1+u^2+v^2-u^2v^2}{(1+uv)^2}$$
$$x^2/t^2= \frac{u^2+2uv+v^2}{(1+uv)^2}$$
$$x^2/t^2= \frac{(u+v)^2}{(1+uv)^2}$$

Success! I guess I just needed to post this thread to figure it out. Thanks for the help.
 
Last edited:
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  • #2
Posting sometimes helps collect your thoughts. In the standard derivation, one would use the Lorentz transformations to find ##dx## and ##dt## in terms of ##dx'## and ##dt'##, then take the ratio.
 

Related to Compute relativistic addition of velocities

1. What is relativistic addition of velocities?

Relativistic addition of velocities is a mathematical formula used to calculate the combined velocity of two objects moving at high speeds, taking into account the effects of special relativity. It is used to determine how fast one object appears to be moving from the perspective of another object in motion.

2. Why is relativistic addition of velocities important?

Relativistic addition of velocities is important because it allows us to accurately calculate the speed of objects moving at high speeds, which is essential for many fields of science, such as astrophysics and particle physics. It also helps us understand the effects of special relativity on the perception of motion.

3. How is relativistic addition of velocities calculated?

The formula for relativistic addition of velocities is v = (u + v) / (1 + uv/c^2), where v is the combined velocity, u is the velocity of the first object, v is the velocity of the second object, and c is the speed of light. This formula takes into account the time dilation and length contraction effects of special relativity.

4. Can relativistic addition of velocities be applied to any speed?

No, relativistic addition of velocities is only applicable to speeds that are a significant fraction of the speed of light. At lower speeds, the effects of special relativity are negligible and the classical formula for adding velocities can be used instead.

5. How does relativistic addition of velocities affect our perception of time and distance?

Relativistic addition of velocities shows that time and distance are not absolute, but are relative to the observer's frame of reference. As an object approaches the speed of light, time appears to slow down and distances appear to contract from the perspective of an outside observer. This is known as time dilation and length contraction, respectively, and is a fundamental aspect of special relativity.

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