How much kinetic energy does a proton gain if it is accelerated, with no friction, through a potential difference of 1.00 V? The proton is 1836 times heavier than an electron

If energy is conserved, then the sum of the potential energy and the kinetic energy is a constant.

Assuming the proton starts from rest, so it's kineitc energy is zero, but it has a potential energy, PE equal to:

PE = qV

where q = 1.6 x 10^-19 C

and V = 1.00 V

Assuming the proton no longer experiences the potential energy and it is all converted to kinetic energy then:

PE * = 0,

KE * = 1 / (2mv^2)

Now since

PE + KE = Total energy = PE * + KE*

Therefore,

qV + 0 = 0 + 1/2mv^2

Or

KE = qV = 1.6 10^-19 J