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Chapter 35: Problem 51

In proton accelerators used to treat cancer patients, protons are accelerated to \(0.61 c\). Determine the energy of the proton, expressing your answer in MeV.

Short Answer

Expert verified
The energy of the proton is approximately 88.12 MeV.

Step by step solution

01

Calculate the relativistic factor (\(\gamma\))

To calculate the relativistic factor, we can use the formula given: $$ \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} $$ Given the velocity \(v = 0.61 c\), we have $$ \gamma = \frac{1}{\sqrt{1 - \frac{(0.61c)^2}{c^2}}} $$
02

Simplify the expression for \(\gamma\) and find its value

Now, we can simplify the expression for the relativistic factor: $$ \gamma = \frac{1}{\sqrt{1 - 0.61^2}} \approx 1.270 $$
03

Calculate the kinetic energy of the proton using the relativistic energy formula

Now that we have the relativistic factor (\(\gamma\)), we can use the relativistic energy formula to calculate the kinetic energy (\(K\)) of the proton: $$ K = (\gamma - 1)mc^2 $$ Using the values for \(\gamma\), the mass of a proton (\(m = 1.67 \times 10^{-27}\) kg), and the speed of light in a vacuum (\(c = 3.00 \times 10^8\) m/s), we get: $$ K = (1.270 - 1)(1.67 \times 10^{-27} \text{ kg})(3.00 \times 10^8 \text{ m/s})^2 $$
04

Calculate the kinetic energy in Joules

Now, perform the calculation to find the kinetic energy in Joules: $$ K \approx 1.412 \times 10^{-13} \text{ J} $$
05

Convert the kinetic energy to MeV

Finally, we need to convert the kinetic energy from Joules to Mega-electron volts (MeV). Since 1 eV = \(1.602 \times 10^{-19}\) J, and 1 MeV = \(10^6\) eV, we can use the conversion factor: $$ \text{MeV} = \frac{\text{J}}{1.602 \times 10^{-19} \text{ J/eV} \times 10^{-6} \text{ MeV/eV}} $$ Substituting the kinetic energy value from step 4, we get: $$ \text{Energy (MeV)} = \frac{1.412 \times 10^{-13} \text{ J}}{1.602 \times 10^{-13} \text{ J/MeV}} \approx 88.12 \text{ MeV} $$ So, the energy of the proton is approximately 88.12 MeV.

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