How Do Relativistic Effects Impact Proton and Electron Velocities?

[ElijahRockers]

Homework Statement

An electron has a speed of .75c. Find the speed of a proton that has:

a) The same kinetic energy as the electron

b) The same momentum as the electron

Homework Equations

γmc² = K + mc²

P = γmv

The Attempt at a Solution

PART A:

m_e (electron mass) = .511MeV/c²
V_e (electron velocity) = .75c

K_e (KE of electron) = (γ-1)m_ec²
K_e = 1.286(.511) = .657MeV

m_p (proton mass) = 938.27MeV/c²
V_p (proton velocity) = ?

So to find V_p:

.657MeV = (γ-1)(938.27MeV) and after simplification, v_p = .0374c

Answer is not in the back, so I can't be sure if I did this correctly.

PART B: (I used the kg representation of mass for this part, because I am confused by the eV units.)

m_e (electron mass) = 9.1E-31 kg
v_e = .75c

P_e (electron momentum) = γm_ev_e
P_e = 4.68E-22 kg*m/s

4.68E-22 kg*m/s = γm_pv_p

I simplified it down to:

2.8E5 = V_p - V_p³/c²

but this looks like a cubic function. I get the feeling that I am doing something wrong.

EDIT: Ok I went back and tried it, this time without plugging any values in, and I arrived at

$V_p = \sqrt{P_{e}^{2} (m_{p}^{2} + \frac{P_{e}^{2}}{c^2})^{-1}}$ and I think that is right. I came up with V_p = 280,239.5 m/s

 

[anathema]

. Is this correct?
Hello, thank you for your post. Your solution for part A looks correct to me. For part B, you are on the right track. The equation you arrived at for finding the velocity of the proton is correct, but you have made a small mistake in your simplification. The correct equation should be:

Vp = 2.8E5 m/s

Remember to take the square root of both sides when isolating for Vp. Great job overall!