# Derivation of the Equation for Relativistic Momentum

• NanakiXIII
In summary, the conversation discusses the reason for the equation for relativistic momentum, p=\gamma mv, and the use of the gamma factor in both the equation for relativistic momentum and the equation for relativistic mass-energy equivalence, E=\gamma mc^2. The conversation also clarifies that relativistic mass is a misnomer and that relativistic momentum is a definition. It is suggested to look into previous threads for leads on derivations for the equation for relativistic momentum.

#### NanakiXIII

I asked a quite similar question about relativistic mass and the reason for this question is identical: I can't seem to dig up any derivation for the equation for relativistic momentum:

$$p=\gamma mv$$

If anyone could point me in the right direction, I'd much appreciate it.

NanakiXIII said:
I asked a quite similar question about relativistic mass and the reason for this question is identical: I can't seem to dig up any derivation for the equation for relativistic momentum:

$$p=\gamma mv$$

If anyone could point me in the right direction, I'd much appreciate it.

Because this is a definition. And so is the relativistic total energy:

$$E=\gamma mc^2$$

I see. Why is it defined as such, exactly? There must be a reason for adding the so frequently seen gamma, no?

Another question on the side: I've seen two different versions of the equation for the mass-energy equivalence, one with and one without the gamma factor. What's the difference?

NanakiXIII said:
I see. Why is it defined as such, exactly? There must be a reason for adding the so frequently seen gamma, no?

I explained that to you in the other thread (relativisic mass) a few minutes ago.

Another question on the side: I've seen two different versions of the equation for the mass-energy equivalence, one with and one without the gamma factor. What's the difference?

Same reason as momentum, I also explained that to you in the other thread.

You mentioned the following:

nakurusil said:
The whole darned thing was introduced in order to reconcile the relativistic momentum/energy:

$$p=\gamma m(0)v$$ (1)
$$E=\gamma m(0)c^2$$

with the Newtonian counterpart:

$$p=mv$$ (2)

So the best thing is to tell your teacher that your proof is you grouped together $$\gamma$$ and proper mass m(0) into $$\gamma m(0)$$ and you assigned that quantity to m

You mean the reason for the gamma is because of the use of relativistic mass? And yet in the other thread you tell me to base relativistic mass on relativistic momentum. That's circular. Are you saying that relativistic mass is a definition and thus relativistic momentum is as well? That would still leave the question of "why the gamma?".

NanakiXIII said:
You mentioned the following:

You mean the reason for the gamma is because of the use of relativistic mass? And yet in the other thread you tell me to base relativistic mass on relativistic momentum. That's circular.

"Relativistic mass" is just a misnomer, ok? Try to learn to forget about it, it has no meaning. Tolman's derivation from the other thread has no "relativistic momentum" in it, ok?

Are you saying that relativistic mass is a definition and thus relativistic momentum is as well? That would still leave the question of "why the gamma?".

Can't you read ? Relativistic momentum is a definition.
"Relativistic mass" is an unfortunate misnomer that corresponds to the result of multiplying $$\gamma$$ by proper mass. It has no physical meaning.

Given the 4-momentum vector $$\tilde P$$,
an inertial observer (with unit 4-velocity $$\tilde t$$
can write that vector as the sum of two vectors, a "temporal" one parallel to $$\tilde t$$ and a "spatial" one perpendicular to $$\tilde t$$.
$$\gamma$$ is $$\cosh{\theta}$$, the analogue of cosine(angle) , and $$\gamma\beta$$ is $$\sinh{\theta}$$, the analogue of sine(angle), where the [Minkowski-]angle $$\theta$$ (called the rapidity) is between $$\tilde P$$ and $$\tilde t$$.

NanakiXIII said:
I can't seem to dig up any derivation for the equation for relativistic momentum:

$$p=\gamma mv$$

(which I should have thought of looking for when I posted in your other thread about relativistic mass)

## 1. What is the equation for relativistic momentum?

The equation for relativistic momentum is p = mv/√(1-v^2/c^2), where p is the momentum, m is the mass, v is the velocity, and c is the speed of light.

## 2. How is this equation derived?

This equation is derived from the relativistic energy equation, E = mc^2/√(1-v^2/c^2), by taking the derivative with respect to velocity. This results in the equation p = mv/√(1-v^2/c^2).

## 3. What does this equation represent?

This equation represents the momentum of an object moving at relativistic speeds, taking into account the effects of special relativity.

## 4. How is this equation different from the classical momentum equation?

The classical momentum equation, p = mv, only applies to objects moving at non-relativistic speeds. The relativistic momentum equation takes into account the changes in mass and velocity at high speeds, resulting in a different equation.

## 5. What are the practical applications of this equation?

This equation is important in fields such as particle physics and astrophysics, where objects are moving at relativistic speeds. It is also used in the development of technologies such as particle accelerators and space travel.