# Show that the total relativistic energy of a proton

• duggielanger
In summary, the mass of a proton when at rest is m. According to an observer using the detector frame, the speed of the anticlockwise moving bunch, A, is such that va^2/c^2=24/25. The total relativistic energy of a proton in bunch A, as observed in the detector frame, is exactly 5mc^2.
duggielanger

## Homework Statement

The mass of a proton when at rest is m. According to an observer using the
detector frame, the speed of the anticlockwise moving bunch, A, is such that
va^2/c^2=24/25
Show that the total relativistic energy of a proton in bunch A, as observed in
the detector frame, is exactly 5mc^2, and work out the speed of the proton,
expressed as a decimal multiple of c, (to 5 significant figures).

## Homework Equations

Right I think its these
E=mc^2
Etot=mc^2/√1-v^2/c^2
Etrans=mc^2/1-v^2/c^2
and maybe
E^2tot=p^2c^2+m^2c^4
p=mv/1-v^2/c^2

## The Attempt at a Solution

Now I know that to get Etot you need e=mc^2 and Etrans and so maybe combing these equations will give the answer but I think this IS not the right to get 5mc^2.
So maybe its E^2tot equation I have to use.
Im just not sure.

well if $\frac{v^{2}}{c^{2}}=\frac{24}{25}$

and the mass of the proton is m, and Etot = mc2$\gamma$

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

then why don't you just try plugging the values in?

I'm not sure what Etrans is supposed to beand to solve for the velocity of the proton all you need is $\frac{v^{2}}{c^{2}}=\frac{24}{25}$

Yeah getting that wrong about Etrans, just looked at my textbooks and was reading it wrong . I have an asnwwer for the last bit not sure if rights but this is what I have for that.
24/25 x 3.00 x10^8 =2.88000 x 10^8 ?

So are you saying just add the values for a proton = 1.67 x 10^-27
the speed of light 3.00 x 10^8
and then add 24=v and 25=c into the gamma part

if $\frac{v^{2}}{c^{2}}=\frac{24}{25}$

then $v=c\sqrt{\frac{24}{25}}$

and just leave it as a multiple of c

for the first part all you need to write is

$E_{tot}=\frac{mc^{2}}{\sqrt{1-\frac{v^{2}}{c^{2}}}}$

and then substitute $\frac{v^{2}}{c^{2}}$ for $\frac{24}{25}$

you should be able to do it in your head, no need to put in the mass of the proton or the exact speed of light, since the answer you want is just 5mc2 where m is the mass of the proton and c is the speed of light

Thank you very much I have now , its taken two days for that to sink in !

## 1. What is the equation for calculating the total relativistic energy of a proton?

The equation is E = mc^2 / (1 - v^2/c^2)^0.5, where E is the energy, m is the mass of the proton, c is the speed of light, and v is the velocity of the proton.

## 2. How does the total relativistic energy of a proton differ from its rest energy?

The total relativistic energy takes into account the kinetic energy of the proton, which is a result of its motion. The rest energy, on the other hand, only accounts for the mass of the proton.

## 3. What is the significance of the speed of light in the equation for the total relativistic energy of a proton?

The speed of light, denoted by 'c', is a fundamental constant in the equation that relates energy and mass. It represents the maximum speed at which any particle in the universe can travel and plays a crucial role in the theory of relativity.

## 4. How does the total relativistic energy of a proton change as its velocity approaches the speed of light?

As the velocity of the proton approaches the speed of light, the denominator of the equation (1 - v^2/c^2)^0.5 becomes smaller and smaller. This means that the total relativistic energy of the proton also becomes larger, approaching infinity as the proton reaches the speed of light.

## 5. How is the concept of relativistic energy important in understanding the behavior of subatomic particles?

Relativistic energy is important in understanding the behavior of subatomic particles because it allows us to accurately calculate their energy and motion, taking into account the effects of special relativity. This is crucial in fields such as particle physics, where high velocities and energies are involved.

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