Answer:
a) [tex] v = c \cdot 0.04 = 1.2\cdot 10^{7} m/s[/tex]
b) [tex] v = c \cdot 0.71 = 2.1\cdot 10^{8} m/s[/tex]
c) [tex] v = c \cdot 0.994 = 2.97\cdot 10^{8} m/s[/tex]
d) [tex] v = c \cdot 0.999 = 2.997\cdot 10^{8} m/s[/tex]
e) [tex] v = c \cdot 0.9999 = 2.999\cdot 10^{8} m/s[/tex]
Explanation:
At that energies, the speed of proton is in the relativistic theory field, so we need to use the relativistic kinetic energy equation.
[tex] KE=mc^{2}(\gamma -1) = mc^{2}(\frac{1}{\sqrt{1-\beta^{2}}} -1)[/tex] (1)
Here β = v/c, when v is the speed of the particle and c is the speed of light in vacuum.
Let's solve (1) for β.
[tex] \beta = \sqrt{1-\frac{1}{\left (\frac{KE}{mc^{2}}+1 \right )^{2}}} [/tex]
We can write the mass of a proton in MeV/c².
[tex] m_{p}=938.28 MeV/c^{2} [/tex]
Now we can calculate the speed in each stage.
a) Cockcroft-Walton (750 keV)
[tex] \beta = \sqrt{1-\frac{1}{\left (\frac{0.75 MeV}{938.28 MeV}+1 \right )^{2}}} [/tex]
[tex] \beta = 0.04 [/tex]
[tex] v = c \cdot 0.04 = 1.2\cdot 10^{7} m/s[/tex]
b) Linac (400 MeV)
[tex] \beta = \sqrt{1-\frac{1}{\left (\frac{400 MeV}{938.28 MeV}+1 \right )^{2}}} [/tex]
[tex] \beta = 0.71 [/tex]
[tex] v = c \cdot 0.71 = 2.1\cdot 10^{8} m/s[/tex]
c) Booster (8 GeV)
[tex] \beta = \sqrt{1-\frac{1}{\left (\frac{8000 MeV}{938.28 MeV}+1 \right )^{2}}} [/tex]
[tex] \beta = 0.994 [/tex]
[tex] v = c \cdot 0.994 = 2.97\cdot 10^{8} m/s[/tex]
d) Main ring or injector (150 Gev)
[tex] \beta = \sqrt{1-\frac{1}{\left (\frac{150000 MeV}{938.28 MeV}+1 \right )^{2}}} [/tex]
[tex] \beta = 0.999 [/tex]
[tex] v = c \cdot 0.999 = 2.997\cdot 10^{8} m/s[/tex]
e) Tevatron (1 TeV)
[tex] \beta = \sqrt{1-\frac{1}{\left (\frac{1000000 MeV}{938.28 MeV}+1 \right )^{2}}} [/tex]
[tex] \beta = 0.9999 [/tex]
[tex] v = c \cdot 0.9999 = 2.999\cdot 10^{8} m/s[/tex]
Have a nice day!
