Final answer:
The speed of the electron with the same momentum and kinetic energy as the proton is approximately 0.8834c.
Step-by-step explanation:
To find the speed of an electron with the same momentum and kinetic energy as a proton moving at a speed of 0.76c, we need to use the formulas for momentum and kinetic energy. The momentum of a particle is given by p = m * v, where p is the momentum, m is the mass, and v is the velocity. The kinetic energy of a particle is given by KE = (1/2) * m * v^2, where KE is the kinetic energy, m is the mass, and v is the velocity.
Since we want the electron to have the same momentum and kinetic energy as the proton, we can set up the equations:
mv = mp * 0.76c → v = (mp * 0.76c) / me
(1/2) * me * v^2 = (1/2) * mp * (0.76c)^2 → v^2 = (mp / me) * (0.76c)^2 → v = (mp / me)^(1/2) * 0.76c
Plugging in the known values for the mass of a proton (mp) and the mass of an electron (me) and performing the calculations, we find that the speed of the electron is approximately 0.8834c. Therefore, the speed of the electron is 1 - 0.8834c = 0.1166c.