SR: Finding the speed of particles wrt the laboratory

In summary: and my article on the geometry of quick relative motion https://www.physicsforums.com/topic/261498-relativity-2-d-motion-geometry-quick-motion/
  • #1
rugerts
153
11

Homework Statement


Two particles in a high-energy accelerator experiment approach each other head-on with a relative speed of 0.890c. Both particles travel at the same speed as measured in the laboratory. What is the speed of each particle, as measured in the laboratory?

Homework Equations


upload_2019-3-22_15-1-21.png
for
upload_2019-3-22_15-1-49.png


The Attempt at a Solution


IMG-1969.JPG

I've interpreted the relative velocity between the two as 0.890c and the velocity of, say particle 2, to be -v wrt laboratory S and -v wrt S'. The answer appears to be 0.611c which is way off from what I've got.
 

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  • #2
After another try, I've made some progress, but it still doesn't seem to be yielding the correct answer. Can anyone offer any helpful hints for my new work shown below:
IMG-1970.JPG
 

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  • #3
So it seems as though using the equation for v prime gives the correct result, 0.611c. Not sure why this is so. My work is shown below... (After the last step shown, I plugged into a Computer Algebra System)
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  • #4
The key here is what the "relative" speed of the two particles means in this problem. After realizing that it is the speed of one particle in the system of reference where the other one is at rest, everything else is straightforward. This system is one which moves at the same speed as one of the particles (for instance, particle 1). Then using the first equation for the motion of the second particle with respect to this system, ##v_x'=-0.89c##, ##u=v##, and ##v_x=-v##.
 
  • #5
quinoa19 said:
The key here is what the "relative" speed of the two particles means in this problem. After realizing that it is the speed of one particle in the system of reference where the other one is at rest, everything else is straightforward. This system is one which moves at the same speed as one of the particles (for instance, particle 1). Then using the first equation for the motion of the second particle with respect to this system, ##v_x'=-0.89c##, ##u=v##, and ##v_x=-v##.
Fair enough. How about using the other equation where the variable solved for is not v' but instead v. Then, rearranging to get v' (this is as opposed to using the equation where v' is the variable solved for). Why doesn't it work both ways?
 
  • #6
Rearranging the second equation and solving for v' will produce the first equation
 
  • #7
Side comment [possibly too advanced for introductory physics... but really shouldn't be]

There is a geometric analogy that can be exploited in this problem.
The relative velocity $$v_{BA}=\tanh(\theta_B-\theta_A)\qquad v_{BA}=0.89,$$
where [itex](\theta_B-\theta_A)[/itex] is the relative Minkowski-angle [called rapidity] between the particles' 4-velocities.

The frame which observes both particles with the same speed in opposite directions is the center of mass frame (technically, center of momentum frame) [assuming they have equal masses]. Call this frame O. The 4-velocity of this frame is along the Minkowski-angle bisector. In that frame, A and B have speed
$$v_{BO}=\tanh(\theta_B-\theta_O)=\tanh\left(\frac{\theta_B-\theta_A}{2}\right)=\tanh\left({\rm arctanh}(0.89)\right)$$
https://www.wolframalpha.com/input/?i=tanh(atanh(0.890)/2) gives 0.61128

(Your first equation, the relative-velocity formula, is actually an identity for the hyperbolic-tangent of a difference of angles:
\begin{align*}
\tanh(\theta_B-\theta_A)&=\frac{\tanh\theta_B-\tanh\theta_A}{1-\tanh\theta_B\tanh\theta_A}\\
\tanh((\theta_B-\theta_O)-(\theta_A-\theta_O))&=\frac{\tanh(\theta_B-\theta_O)-\tanh(\theta_A-\theta_O)}{1-\tanh(\theta_B-\theta_O)\tanh(\theta_A-\theta_O)}\\
\end{align*}
.)

For fancier approaches: see my Insight https://www.physicsforums.com/insights/relativity-variables-velocity-doppler-bondi-k-rapidity/
 
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1. How do you find the speed of particles with respect to the laboratory in special relativity?

In special relativity, the speed of particles with respect to the laboratory can be found using the Lorentz transformation equations. These equations take into account the relative velocity between the particle and the laboratory, as well as the speed of light, to determine the particle's velocity in the laboratory frame of reference.

2. Why is it important to calculate the speed of particles with respect to the laboratory in special relativity?

Calculating the speed of particles with respect to the laboratory is important in special relativity because it allows us to understand how time, length, and mass are affected by the relative motion between the particle and the observer. This is crucial for accurately describing the behavior of particles at high speeds.

3. What factors affect the speed of particles with respect to the laboratory in special relativity?

In special relativity, the speed of particles with respect to the laboratory is affected by the relative velocity between the particle and the observer, as well as the speed of light. Additionally, the particle's initial velocity and the observer's frame of reference can also impact the calculated speed.

4. Can the speed of particles with respect to the laboratory exceed the speed of light in special relativity?

No, according to special relativity, the speed of particles with respect to the laboratory cannot exceed the speed of light. This is a fundamental principle of the theory and is supported by experimental evidence. As an object approaches the speed of light, its mass and energy increase, making it impossible to reach or exceed the speed of light.

5. How does the concept of time dilation in special relativity affect the calculation of the speed of particles with respect to the laboratory?

The concept of time dilation in special relativity states that time appears to slow down for an observer moving at high speeds compared to a stationary observer. This can affect the calculation of the speed of particles with respect to the laboratory because the time measured by the observer may be different from the time experienced by the moving particle, leading to different calculated speeds.

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