# Understanding the Derivation of Relativistic Mass in Inelastic Collisions

• mainguy
In summary: So, according to your diagram, in frame ##S## the particle at point ##A## will have its velocity doubled, from ##v## to ##u##. In frame ##S^-1##, the particle at point ##B## will have its velocity tripled, from ##v## to ##u^2##. This seems to me to violate the law of conservation of momentum, since momentum is a conserved quantity.
mainguy
Hi guys, thanks for helping with this! I'm a little stuck with this question about the derivation for relativistic mass.

1. Homework Statement

By considering the inelastic collision of two balls as perceived in different reference frames show that the relativistic mass is equal to the rest mass multiplied by the gamma factor (sqrt(1-u^2/c^2).

## Homework Equations

So I know the factor is 1/(1-u2/c2) but proving it is tough.
I've considered a reference frame moving at u in a direction perpendicular to the collision, so basically this:

## The Attempt at a Solution

The vertical velocity u0 is transformed by a gamma factor, u = u'*gamma
So it slows down slightly as expected in the moving frame

It seems to me that the ball moving sidelong, say B in the first image, will have it's velocity altered in two parts. The vertical component will be multiplied by a gamma factor, and the horizontal component will transform as a lorentz:
u'= (u -v)/(1-uv/c2)

It seems clear to me that the vertical velocities of A and B are identical, and that they transform in an identical manner.

From class I know this isn't true, apparently they are identical velocities but they transform in a different manner. But I don't see how B could be transformed via the Lorentz formula if only a compnent of it's velocity is along the line parallel to the motion of the moving frame. Help would be much appreciated![/B]

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mainguy said:
Hi guys, thanks for helping with this! I'm a little stuck with this question about the derivation for relativistic mass.

1. Homework Statement

By considering the inelastic collision of two balls as perceived in different reference frames show that the relativistic mass is equal to the rest mass multiplied by the gamma factor (sqrt(1-u^2/c^2).

## Homework Equations

So I know the factor is 1/(1-u2/c2) but proving it is tough.
I've considered a reference frame moving at u in a direction perpendicular to the collision, so basically this:

View attachment 221980

## The Attempt at a Solution

The vertical velocity u0 is transformed by a gamma factor, u = u'*gamma
So it slows down slightly as expected in the moving frame

It seems to me that the ball moving sidelong, say B in the first image, will have it's velocity altered in two parts. The vertical component will be multiplied by a gamma factor, and the horizontal component will transform as a lorentz:
u'= (u -v)/(1-uv/c2)

It seems clear to me that the vertical velocities of A and B are identical, and that they transform in an identical manner.

From class I know this isn't true, apparently they are identical velocities but they transform in a different manner. But I don't see how B could be transformed via the Lorentz formula if only a compnent of it's velocity is along the line parallel to the motion of the moving frame. Help would be much appreciated![/B]

This is a problem that was treated by Planck in a 1906 paper, but the arguments there are not particularly enlightening. A much nicer and more convincing argument was advanced in a 1909 paper by Lewis and Tolman. You can find this argument nicely laid out more-or-less completely on pages 48--50 of "A History of the Theories of Aether and Electricity, Volume II", by Sir Edmund Whittaker, Harper (1953).

According tot he diagram, frame ##S## is moving to the right at speed ##V##, but according to what you wrote, you have it moving vertically at speed ##u##. You also use ##u## for the speed of one of the particles in the collision.

## What is relativistic mass?

Relativistic mass is a concept in physics that describes the increase in mass of an object as it approaches the speed of light. It is a result of the relationship between mass and energy in Einstein's theory of relativity.

## How is relativistic mass calculated?

The formula for calculating relativistic mass is m = m0/√(1-v^2/c^2), where m0 is the rest mass of the object, v is its velocity, and c is the speed of light. This formula takes into account the increase in mass due to the object's high velocity.

## What is the significance of relativistic mass?

The concept of relativistic mass is significant because it helps to explain the behavior of objects at high speeds, such as those near the speed of light. It also provides a deeper understanding of the relationship between mass and energy.

## How does relativistic mass differ from rest mass?

Rest mass is the mass of an object when it is at rest, while relativistic mass takes into account the object's velocity. Rest mass is constant, but relativistic mass increases as the object's velocity increases.

## Is relativistic mass a real or apparent phenomenon?

Relativistic mass is a real phenomenon that has been confirmed through experiments and observations. It is a fundamental concept in Einstein's theory of relativity and is crucial in understanding the behavior of objects at high speeds.

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