Welcome to PF twinklestar28!
twinklestar28 said:
Homework Statement
Write the momentum four vector of a particle that has:
Rest mass = 80 Gev/c^2
Energy = 100 Gev
and moves along the z axis.What is its speed as a fraction of the speed of light.
Homework Equations
E^2 = p^2c^2 + m^2C^4
The Attempt at a Solution
Momentum four vector - (0, 0, (3*10^10)v, 1.1*10^15) in units of Gev/c
The speed i got was v/c = 1.22*10^-34 but when I double check it, it doesn't seem to match. I'm not sure where I'm going wrong, is my four momentum vector wrong?
Your four-vector confuses me a bit. I'm used to seeing the timelike component listed first, but I am assuming that you are using a convention where the timelike component of the vector is the fourth and last one. In any case, using that convention the momentum four-vector is just given by P = (p
x, p
y, p
z, E/c), where E is the relativistic energy, and (p
x, p
y, p
z) are the components of the three-momentum, p. In this case, we have p
x = p
y = 0, and p = p
z = γmv, where v is the velocity (which is entirely in the z-direction). So your four-vector becomes
P = (0, 0, γv(80 GeV/c
2), (100 GeV/c) ).
There is no sense converting anything into SI units in any part of this problem. I suspect that's the problem you're having with calculating v/c: the unit conversions. I recommend leaving everything in terms of GeV and just carrying the various factors of c along for the ride in the calculation. In fact, most physicists would use a unit system where c = 1, and everything (mass, momentum, energy) is measured in eV. The pesky factors of c would then disappear from the calculation entirely. In fact, we could write E
2 = p
2 + m
2 in that unit system. But for now, I'll keep all the pesky factors of c in there, to avoid confusing you (I hope). To calculate v/c, start with the equation that you wrote for the relativistic energy:$$E^2 = p^2c^2 + m^2c^4$$Solve for p:$$pc = \sqrt{E^2 - m^2c^4} = \sqrt{(100~\textrm{GeV})^2 - (80~\textrm{GeV}/c^2)^2 \cdot c^4}$$Notice that the c's cancel entirely from the second term under the square root, just leaving us with:$$pc = \sqrt{(100~\textrm{GeV})^2 - (80~\textrm{GeV})^2} = 60~\textrm{GeV}$$
So, p = (60 GeV/c). Using this, and the fact that p = γmv, you should be able to solve for v/c, which in turn tells you γ. Again, DON"T plug in values for c in SI units. Just carry the factors of c along for the ride in your algebra.
