Question

An electron is accelerated from rest through a potential difference of 1.40*10^5 V. What is its speed?

So there were also two parts before to this question: the relativistic kinetic energy in eV (which I correctly found to be just 1.40*10^5 eV) and its relativistic total energy in eV (which I correctly found to be 651000 eV).

Anyways, to find the speed of the electron, I used:

Total E = (gamma)(m)(c)^2
Total E = (1/sqrt(1-(v/c)^2))(m)(c)^2

I rearranged that to isolate v:

v = sqrt(c^2 - ((mc^4)/TotalE)^2)

c = 3.00*10^8 m/s
m = 0.511*10^6 eV/c^2
TotalE = 651000eV

So then v = sqrt(3.00*10^8)^2 - ((0.511*10^6)(3.00*10^8)^4)/651000)^2

But I get a (-) value within the square root brackets, more specifically I get -4.990738814*10^33 which is where I am not stuck. By the way, my answer key says that the answer I am supposed to get is v = 0.62c. Anything I did wrong?

Answer 1

Sorry, I meant to say towards the end "more specifically [...] which is where I AM stuck"

Answer 2

v = c*sqrt(1 - ((0.511*10^6)(3.00*10^8m/s)^2)/651000)^2
now examine the units mass in particular
=c*sqrt(1 - ((0.511*10^6ev/c^2)*(c)^2)/651000ev)^2
=c*sqrt(1 - ((0.511*10^6ev/c^2)*(c)^2)/651000ev)^2
=c*sqrt(1 - ((0.511*10^6ev)^2)/651000ev)^2
=c(sqrt(1-(.511e6/65100)^2)
check my thinking.

Answer 3

I get .619c
6.51 = 5.11/sqrt(1 -beta^2)

sqrt(1-beta^2) = .785

1-beta^2 = .616

beta^2 = .384

beta = v/c = .6195 or v = .62 c