How do I calculate the mass of a moving proton?

Question

When a proton moves it's no longer its rest mass, it now has more mass because of the Kinetic Energy. So if we're to find the new protons mass if it was moving at some speed, how would we find it? Would we convert the kinetic energy to mass then plus that on the rest mass? Or is it just Kinetic Energy + Rest mass?

Answer 1

Suppose we accelerate a proton to a speed $v$ then crash it into some large mass $M$ so that the proton comes to rest inside the large mass. Then we measure the speed $u$ that the large mass + proton are moving. We'll assume the final speed $u \ll c$ so that regular Newtonian mechanics applies.

We know conservation of momentum always applies, and the momentum of the large mass + proton is $p_M = (M+m_p)u$, and this has to be equal to the initial momentum of the proton $p_p$. So now we know the momentum that the proton had when it was moving at the initial speed $v$.

If we do this experiment we will find that the momentum of the proton moving at speed $v$ was:

$$ p_p = \gamma m_p v \tag{1} $$

where $m_p$ is the rest mass of the proton and $\gamma$ is the Lorentz factor:

$$ \gamma = \frac{1}{\sqrt{1 - v^2/c^2}} $$

Now there are two ways of looking at equation (1). One option is to define a relativistic mass as:

$$ m_r = \gamma m_p \tag{2} $$

Then the equation for the momentum becomes $p = m_r v$ and this looks just like the Newtonian equation for the momentum that we all learned at school. If you take this option then equation (2) defines the mass of the moving proton as you asked.

But I strongly recommend you do not do this as the relativistic mass is not a concept used in modern physics and you will not be taught it if you do a physics degree. For more on this see Why is there a controversy on whether mass increases with speed?

Instead we take the term mass of the proton to only ever mean its rest mass, and we accept that the Newtonian expression for the momentum is an approximation that needs to be modified to equation (1) for speeds comparable to the speed of light.

Since you mentioned the energy, we can show that the total energy of a moving object with rest mass is given by:

$$ E = \gamma mc^2 $$

Thus includes the rest energy given by Einstein's famous equation $E = mc^2$, so to get the kinetic energy we subtract off $mc^2$ to get the kinetic energy:

$$ KE = (\gamma - 1)mc^2 $$

At speeds much less than $c$ this is approximately equal to the Newtonian expression:

$$ KE = \tfrac12 mv^2 $$

Answer 2

Relativistic mass $= m_{rest}+ \frac{E_{kin}}{c^2}$

Relativistic mass$ = \frac{1}{\sqrt{ 1-v^2/c^2 }} * m_{rest}$

Relativistic mass of proton is good for:

1. Calculating force needed to steer the proton (to change its direction while not changing its speed)

2. Calculating the rest mass of a box with a moving proton inside.