Relativity- CM and Lab frame energy

• trelek2
In summary, the conversation revolves around trying to solve two relativity past paper questions related to an elastic collision of two electrons at relativistic speeds. The first question involves showing that the total energy of the electron in motion in the lab frame can be expressed as E1= (2E*^2 - m^2c^4)/mc^2, where E* is the energy of one of the electrons in the center of mass frame. The second question asks for the scattering angle in the lab frame, given that it is 90 degrees in the center of mass frame, and involves using the formula for E1. The conversation also mentions various attempted solutions using different equations, but the solution remains elusive.
trelek2

Homework Statement

Hi, I've been doing some relativity past paper questions and I totally got stuck on 2 of them.
Both regarding an elastic collision of 2 electrons at relativistic speeds.

First question: I am to show that it is true that the total energy of the electron which is in motion in the lab frame can be given by the following formula:
$$E _{1}= \frac{2E* ^{2}-m ^{2} c ^{4} }{mc ^{2} }$$

Where E* is the energy of one of the electrons in the CM frame.
E is energy in lab frame, m is mass, c is speed of light

Second question: According to an observer in CM frame scattering angle is 90 degrees.
Whais is the scattering angle in Lab frame (in terms of E* and m)

2. The attempt at a solution

I tried doing it in various ways and I can't get an answer.

My 2 best bets are noticing that:
$$E _{1}= mc ^{2} + \frac{2p ^{*2} _{1}}{m }$$
where p* is the momentum of an electron in CM frame, but what next??

Or I try this way, don't even know if its valid:
(total energy in Lab frame) squared=(total energy in CM frame) squared
So for lab frame total energy is
$$E _{1} +mc ^{2}$$
and for CM frame i take 2E*
I can then expand any of these E's taking
$$E = \sqrt{m ^{2} c ^{4}+(pc) ^{2} }$$
This gets me really close to the answer but I have an additional term i can't get rid of: $$\frac{p _{1} ^{2}c ^{2} }{2}$$, which is the momentum of moving particle in lab frame times c^2

, I assume I have to use the formula for E1 but don't know where to go from there.Any help would be much appreciated!

1. What is the difference between the CM frame and the Lab frame in terms of energy?

The CM (center of mass) frame is a reference frame where the total momentum of a system is equal to zero. In this frame, the total energy of the system is equal to the rest mass energy of the particles involved. The Lab frame, on the other hand, is an external frame of reference where the system is observed to have a non-zero total momentum. In this frame, the total energy of the system is greater than the rest mass energy due to the additional kinetic energy of the particles.

2. How does special relativity explain the relationship between energy and mass?

In special relativity, energy and mass are interchangeable and are both components of the same physical quantity known as the energy-momentum 4-vector. This means that as an object gains energy, its mass also increases. This is described by Einstein's famous equation, E=mc², where E is energy, m is mass, and c is the speed of light.

3. How does the energy of a particle change when it moves at relativistic speeds?

As a particle approaches the speed of light, its energy increases exponentially. This is because, according to special relativity, an object's energy is equal to its rest mass energy multiplied by the Lorentz factor, which increases as the object's velocity approaches the speed of light. This means that a small increase in velocity can result in a large increase in energy for a relativistic particle.

4. Can the total energy of a system be conserved in both the CM and Lab frames?

Yes, the total energy of a system is conserved in both the CM and Lab frames. In the CM frame, the total energy is equal to the rest mass energy of the particles, while in the Lab frame, the total energy is equal to the sum of the rest mass energies and the kinetic energies of the particles. However, the distribution of energy between kinetic and potential energy may differ in the two frames.

5. How does the concept of time dilation affect the energy of a particle in motion?

According to special relativity, time is relative and is affected by an object's velocity. As an object's velocity increases, time appears to slow down for that object. This means that a particle in motion will have a lower rate of energy consumption compared to an identical particle at rest, as observed from an external frame. However, the total energy of the particle remains the same in both frames.

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