# How Does a Nucleus's Mass Change After Emitting a Gamma-Ray Photon?

• Pedro de la Torre
The energy and momentum conservation equations hold for the 4-vectors, not just the magnitudes. So, in your calculation, when you say "Pg = -Pγ", you are taking the 3-vectors and equating them, but you are not equating the 4-vectors. In general, if ##p## is the 3-vector momentum, then ##p = (E/c, \vec{p})## is the 4-vector momentum. So, the 4-momentum conservation equation is ##P_i = P_f##, where the subscript "i" stands for "initial" and "f" stands for "final".In summary, a stationary excited nucleus decays to its
Pedro de la Torre

## Homework Statement

A stationary excited nucleus decays to its grpund state by emitting a γ-ray photon of energy Eγ. The grpund state nucleus recoils in the opposite direction at speed v. Show that when v<<c the change of mass of the nucleus is approximately:

me - mg = Eγ/c2 [1+ 1/2(v/c)]

## Homework Equations

v/c=pc/E
PμPμ = constant = (mc2)2
E2 = (mc2)2 + (pc)2
Pγc = Eγ

## The Attempt at a Solution

I used both, the cuadrimomentum conservation and the energy conservation. I think the solution should be directly found from the cuadrimomentum conservation, but I can not solve it.

Pedro de la Torre said:

## The Attempt at a Solution

I used both, the cuadrimomentum conservation and the energy conservation.
Conservation of 4-momentum implies conservation of both relativistic energy and relativistic 3-momentum.

I think the solution should be directly found from the cuadrimomentum conservation, but I can not solve it.
I'm not following what you mean here. It would be helpful if you showed some detail of your attempt to solve it this way.

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Your work looks good, except I don't understand why you suddenly put in a negative sign at the very end. ##\frac{E_{\gamma}}{E_g}## is positive. ##E_{\gamma} = |p_{\gamma}|c##.
So, ##1 + \frac{E_{\gamma}}{2 E_g} = 1+ \frac{v}{2c}##.

Last edited:
I did that because I considered there Pg = - Pγ. But I think I am realising that this is only for the vectorial form, Isn´t it?

Pedro de la Torre said:
I did that because I considered there Pg = - Pγ. But I think I am realising that this is only for the vectorial form, Isn´t it?
Yes.

## 1. What is relativistic dynamics?

Relativistic dynamics is a branch of physics that studies the motion of objects at speeds approaching the speed of light. It combines principles of classical mechanics and special relativity to explain the behavior of particles moving at high speeds.

## 2. What is the difference between classical and relativistic dynamics?

Classical dynamics, also known as Newtonian dynamics, describes the motion of objects at low speeds. Relativistic dynamics, on the other hand, accounts for the effects of special relativity at high speeds, such as time dilation and length contraction.

## 3. What is the equation for relativistic dynamics?

The equation for relativistic dynamics is F = dp/dt, where F is the force applied to an object, p is the object's momentum, and t is time. This equation takes into account the relativistic mass of the object, which increases as its speed approaches the speed of light.

## 4. How does relativistic dynamics impact space travel?

Relativistic dynamics plays a crucial role in space travel as it explains the behavior of objects moving at high speeds, such as spacecrafts. It also helps calculate the effects of time dilation and length contraction on space travel, making it possible to accurately plan and execute missions.

## 5. What is an example of a relativistic dynamics problem?

An example of a relativistic dynamics problem is calculating the force required to accelerate a spaceship to a certain speed, taking into account the increase in relativistic mass at high speeds. This problem would also involve considering the effects of time dilation on the journey.

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