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?
Sorry, I meant to say towards the end "more specifically [...] which is where I AM stuck"
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.
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
It looks like you made a mistake in your calculations. Let's go through the steps one by one to find the error.
Starting from the equation you mentioned:
v = sqrt(c^2 - ((mc^4)/TotalE)^2)
We know:
c = 3.00*10^8 m/s
m = 0.511*10^6 eV/c^2
TotalE = 651000 eV
First, let's convert the mass from eV/c^2 to kilograms, since we are using SI units.
m = (0.511*10^6 eV/c^2) * (1.60218*10^-19 J/eV) / (8.9875*10^16 J/c^2) = 9.10938356*10^-31 kg
Now, substitute the values into the equation:
v = sqrt((3.00*10^8 m/s)^2 - ((9.10938356*10^-31 kg) * (3.00*10^8 m/s)^4) / (651000 eV))^2)
v = sqrt(9.00*10^16 m^2/s^2 - (9.10938356*10^-31 kg) * (3.00*10^8 m/s)^4 / (651000 eV))^2)
Now, let's calculate the value inside the square root brackets:
v = sqrt(9.00*10^16 m^2/s^2 - (9.10938356*10^-31 kg) * (3.00*10^8 m/s)^4 / (651000 eV))^2)
v = sqrt(9.00*10^16 m^2/s^2 - (9.10938356*10^-31 kg) * (3.00*10^8 m/s)^4 / (651000 eV))^2)
v = sqrt(9.00*10^16 m^2/s^2 - 2.2975*10^15 m^2/s^2)^2
Now simplify the expression inside the square root:
v = sqrt(6.67025*10^16 m^2/s^2)^2
v = sqrt(6.67025*10^16 m^2/s^2)
v = 8.167 m/s
Therefore, the speed of the electron is 8.167 m/s, which is approximately 0.0272c (where c is the speed of light). This is different from the answer you provided (0.62c). It's possible that there was an error in the calculations or a different method was used. Double-check your calculations and units to ensure accuracy.