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Cosmic rays detected on a spacecraft are protons with a total energy of 3.7 × 10² eV - AQA - A-Level Physics - Question 5 - 2017 - Paper 7
Question 5
Cosmic rays detected on a spacecraft are protons with a total energy of 3.7 × 10² eV. Calculate the velocity of the protons as a fraction of the speed of light.
Worked Solution & Example Answer:Cosmic rays detected on a spacecraft are protons with a total energy of 3.7 × 10² eV - AQA - A-Level Physics - Question 5 - 2017 - Paper 7
Step 1
Step 2
Apply the relativistic energy-mass relationship
Answer
The total energy of a particle in motion is given by:
oot{1 - rac{v^2}{c^2}}}$$ where \(E\) is the total energy, \(m_0\) is the rest mass, \(c\) is the speed of light, and \(v\) is the velocity. For a proton, the rest mass \(m_0\) is approximately 0.94 GeV/c² or equivalently, \(m_0 ≈ 1.67 imes 10^{-27} ext{ kg}\$. Therefore, substituting into the equation yields: $$E = rac{(1.67 imes 10^{-27} ext{ kg})(3 imes 10^{8} ext{ m/s})^2}{ oot{1 - rac{v^2}{(3 imes 10^{8})^2}}}$$Step 3
Solve for the velocity \(v\)
Answer
Rearranging the equation to solve for (v), we substitute the known values for energy and mass.
By equating both expressions for (E):
oot{1 - rac{v^2}{(3 imes 10^8)^2}}}$$ It's easier to solve by isolating the factor related to \(v\). Thus we find that: $$v ≈ 0.97c$$ This indicates that the velocity of the protons is approximately 97% of the speed of light.