# Lorentz Velocity Addition Problem

• JoeyBob
In summary: Indeed, you don't even need to calculate the corresponding x-component in the primed frame.However, you will need to use values of x and t (or x' and t') to calculate the value of y or y' for a particular instant in time.So, for the first part of the problem, you should use the formula for the y-component of velocity transformation, which you can also find in the link above. Use the values given in the problem to calculate the y-component of velocity in the primed frame.However, as I said above, if you are given the x-component and the y-component of velocity in the unprimed frame, you can calculate the corresponding x-component and the y-component in the primed frame
JoeyBob
Homework Statement
see attached
Relevant Equations
u'=u/(δ(1-(uv)/c^2)
So for the formula, u'=u/(δ(1-(uv)/c^2)

u=2.06E8 and v=0. I am only looking at the y components here.

Since v=0 it really becomes u'=u/δ or u'= u*sqrt(1-(u^2)/c^2)).

Anyways when I plug this in I am getting 1.49E8 when the answer should be 0.951E8. Am I not using the correct formula here?

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There are three velocities in this problem. Are you sure you're using the right one when calculating ##\gamma##?

vela said:
There are three velocities in this problem. Are you sure you're using the right one when calculating ##\gamma##?
Its asking for the y component, which is j hat. The particle has a y-component velocity while the moving observer does not.

Make sure you are taking care of the x direction relative velocity first; then you can work on the y direction relative velocity. You will need to perform relative velocity calculation twice. Keep in mind, relative motion has an effect on both time (indenpdent of direction) and space (dependent of direction).

guv said:
Make sure you are taking care of the x direction relative velocity first; then you can work on the y direction relative velocity. You will need to perform relative velocity calculation twice. Keep in mind, relative motion has an effect on both time (indenpdent of direction) and space (dependent of direction).
Why would I need to do x component first? Couldnt i just ignore x component and do y component since that's what the question is asking?

You're trying to find ##\vec v##, the velocity of the particle in the the rest frame of A. So why are you using the formula to find ##v_y'##? Also, you need to understand what the variables in the formula stand for. From your original attempt, it seems you have a misunderstanding there.

PeroK
vela said:
You're trying to find ##\vec v##, the velocity of the particle in the the rest frame of A. So why are you using the formula to find ##v_y'##? Also, you need to understand what the variables in the formula stand for. From your original attempt, it seems you have a misunderstanding there.

u'=u/(δ(1-(uv)/c^2)

u' is the velocity of the object according to the individual moving at velocity v.

u is the velocity of the object according the the individual in the rest frame.

The question is asking for u`', the velocity of the object according to the individual moving at velocity v.

Am I incorrect anywhere in the above?

Except both individuals are at rest if we only look at the y-component.

It's a little confusing because you're mixing notation and reusing variables. According to A', the particle moves with velocity ##\vec v'##, so the usual convention is to say the particle moves with velocity ##\vec v## according to A.

JoeyBob said:
Except both individuals are at rest if we only look at the y-component.
That's not how it works.

vela said:
That's not how it works.
How does it work then? I've determined the x-component of the velocity to be 2.7784E8 using ux'=(ux-v)/(1-(ux*v)/c^2) where I defined the variables in the same manner. i believe my answer is correct, though I have no way of veryfying.

An earlier comment said I needed to find the x-component first in order to find the y-component. The vital part I am missing is why that is the case.

You said
$$u_y'= \frac{u_y}{\gamma[1-(u_y v)/c^2]}.$$ If you write ##\gamma## in terms of ##v##, you have
$$u_y'= \frac{u_y\sqrt{1-(v/c)^2}}{1-(u_y v)/c^2}.$$ You set ##v=0## in the denominator but not in the numerator. Hopefully, it's obvious why you can't do that.

vela said:
You said
$$u_y'= \frac{u_y}{\gamma[1-(u_y v)/c^2]}.$$ If you write ##\gamma## in terms of ##v##, you have
$$u_y'= \frac{u_y\sqrt{1-(v/c)^2}}{1-(u_y v)/c^2}.$$ You set ##v=0## in the denominator but not in the numerator. Hopefully, it's obvious why you can't do that.
But if I do both numerator/denominator u'=u, which isn't the right answer.

JoeyBob said:
But if I do both numerator/denominator u'=u, which isn't the right answer.
First, you can find the velocity transformation equations here (summarised in the light blue box):

https://en.wikipedia.org/wiki/Velocity-addition_formula#Special_relativity

Assuming you understand or accept these formulas, then all you have to do is get the variables correct.

If you are asked only for the y-component of velocity in the unprimed frame, then you don't need to calculate the x-component.

## What is the Lorentz Velocity Addition Problem?

The Lorentz Velocity Addition Problem is a problem in special relativity that arises when trying to add velocities in different reference frames. It is based on the theory of relativity, which states that the laws of physics are the same for all observers in uniform motion.

## Why is the Lorentz Velocity Addition Problem important?

The Lorentz Velocity Addition Problem is important because it helps us understand how velocities behave in different reference frames and how the laws of physics are affected by relative motion. It is also essential for accurately predicting the behavior of objects traveling at high speeds, such as particles in particle accelerators.

## How is the Lorentz Velocity Addition Problem solved?

The Lorentz Velocity Addition Problem is solved using the Lorentz transformation equations, which relate the space and time coordinates of events in one reference frame to those in another reference frame. These equations take into account the effects of time dilation and length contraction.

## What is the formula for solving the Lorentz Velocity Addition Problem?

The formula for solving the Lorentz Velocity Addition Problem is v = (u + v') / (1 + (uv'/c^2)), where v is the velocity of an object in one reference frame, u is the velocity of that reference frame relative to another, and v' is the velocity of the object in the other reference frame.

## What are some real-world applications of the Lorentz Velocity Addition Problem?

The Lorentz Velocity Addition Problem has many real-world applications, such as in particle accelerators, where particles are accelerated to high speeds and their behavior is predicted using the principles of special relativity. It is also important in astronomy, as it helps us understand the behavior of objects moving at high speeds, such as stars and galaxies. Additionally, it has practical applications in GPS technology and satellite communication systems.

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