Final answer:
To calculate the speed of a proton after it accelerates through a potential difference of 215 V, use the formula Kf - Ki = qV. Both the proton and the electron will have the same resulting speed due to their opposite charges. The speed of the proton is approximately 1.56 x 10^6 m/s.
Step-by-step explanation:
To calculate the speed of a proton after it accelerates through a potential difference of 215 V, we can use the formula for the change in kinetic energy:
Kf - Ki = qV
where Kf is the final kinetic energy, Ki is the initial kinetic energy (which is 0 for both the proton and the electron), q is the charge of the particle, and V is the potential difference. Since both the proton and the electron have the same charge magnitude but opposite signs, the resulting speeds will be the same.
Using the given potential difference of 215 V and the charge of a proton (q = 1.60 x 10^-19 C), we can solve for the final kinetic energy:
Kf = qV = (1.60 x 10^-19 C)(215 V) = 3.44 x 10^-17 J
Finally, we can use the equation for kinetic energy to find the speed:
K = (1/2)mv^2
where K is the kinetic energy, m is the mass of the particle, and v is the speed. Rearranging the equation, we can solve for v:
v = sqrt((2K)/m) = sqrt((2(3.44 x 10^-17 J))/(1.67 x 10^-27 kg)) ≈ 1.56 x 10^6 m/s