# Special relativity: find speed and kinetic energy

1. Apr 28, 2010

### k77i

1. The problem statement, all variables and given/known data

A proton (rest mass 1.67x10^-27 kg) has total energy that is 3.2 times its rest energy. What is:
a) the kinetic energy of the proton (in joules)
b) the magnitude of the momentum of the proton (in kg*m/s)
c) the speed of the proton (in terms of the speed of light "c")

2. Relevant equations

E(0) = m(0)c^2
where E(0) is rest energy, m(0) is rest mass and c is the speed of light (approximately 3.00x10^8 m/s)

E = mc^2
where E is the total energy (im not too sure if thats the formula for total energy) and m is the relativistic mass

p=mv
where p is the momentum

E^2 = (p^2)(c^2) + (m(0)^2)(c^2)

Ke = E - m(0)c^2
where Ke is the kinetic energy

3. The attempt at a solution

I found the rest energy E(0) = m(0)c^2 = 1.503x10^-10 and since my total energy = 3.2 times the rest energy, E = 3.2E(0) = 4.81x10^-10.
And because E = mc^2, i can use the previously calculated value to find m which gave me m= 5.344x10^-27.
I used these values and the equation E^2 = (p^2)(c^2) + (m(0)^2)(c^2)to find the momentum which gave me 1.603x10^-18, which is correct.

But when i use the formula p = mv and rearrange it as v = p/m, i get 0.9999c m/s which according to my homework isn't correct.

And also, when i use the equation Ke = E - m(0)c^2 to find the kinetic energy, i get 3.307x10^-10, which is also incorrect.

I know it's long but a little help would be appreciated.

2. Apr 28, 2010

### espen180

Relativistic momentum is $$p=\gamma m_0v$$ and relativistic kinetic energy is $$(\gamma -1)m_0c^2$$ with $$\gamma=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$$

See if you get the right result with these expressions.

3. Apr 28, 2010

### k77i

but how would i calculate the gamma value if i don't have the velocity?

4. Apr 28, 2010

### espen180

Total energy is $$E=\gamma m_0c^2$$ and rest energy is $$E_0=m_0c^2$$. Now take another look at the problem statement. Can you figure out what the velocity and the value of gamma is?

5. Apr 28, 2010

### k77i

Ok then suppose that before i look for the velocity, i want to find the kinetic energy first. If i use the equations E= (gamma)m(0)c^2 and E(0) = m(0)c^2 , i can find the value for gamma, which gives me 3.2. if i then use the equation (gamma -1)m(0)c^2 to find the kinetic erergy it gives me 3.307x10^-10. Why is this answer wrong?