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Chapter 20: Problem 40

Consider the following, all moving in free space: a \(2.0\) -eV photon, a \(0.40-\mathrm{MeV}\) electron, and a \(10-\mathrm{MeV}\) proton. \((a)\) Which is moving the fastest? (b) The slowest? ( \(c\) ) Which has the greatest momentum? \((d)\) The least? (Note: A photon is a light particle of zero mass.)

Short Answer

Expert verified
(a) The photon is moving the fastest, at the speed of light \(3.0 \times 10^8 m/s\). (b) The proton is moving the slowest with speed \(7.73 \times 10^7 m/s\). (c) The proton has the greatest momentum at \(1.29 \times 10^{-19} kg \, m/s\). (d) The photon has the least momentum at \(6.64 \times 10^{-34} kg \, m/s\).

Step by step solution

01

Calculate the Speed of the Photon

All photons move at the speed of light, regardless of their energy. Therefore, the photon's speed is \(3.0 \times 10^8 \, m/s\).
02

Calculate the Speed of the Electron

For a \(0.40-MeV\) electron, first convert the energy from electron volts to joules using the conversion \(1eV = 1.6 \times 10^{-19}J\). The energy in joules is \(0.40 \times 10^6 eV \times 1.6 \times 10^{-19} j/eV = 6.4\times10^{-14}J\). To find the speed, we can use the formula \(E = \frac{1}{2}mv^2\), where \(E\) is the energy, \(m\) is the mass of the electron, and \(v\) is its speed. Rearranging the formula gives \(v = \sqrt{\frac{2E}{m}}\). By substituting the values for the electron's energy and rest mass \(m=9.1\times10^{-31}kg\), we get \(v = \sqrt{\frac{2 \times 6.4\times10^{-14} J}{9.1\times10^{-31} kg}} = 1.18 \times 10^8 \, m/s\).
03

Calculate the Speed of the Proton

Carrying out a similar process for the proton, the kinetic energy of a proton is given as \(10MeV = 10 \times 10^6 eV \times 1.6 \times 10^{-19} J/eV = 1.6 \times 10^{-12}J\). Using the rest mass of a proton \(m = 1.67 \times 10^{-27}kg\), we get the speed \(v = \sqrt{\frac{2 \times 1.6 \times 10^{-12} J}{1.67 \times 10^{-27} kg}} = 7.73 \times 10^7 \, m/s\).
04

Compare the Speeds

Comparing the speeds, the photon will always be the fastest at the speed of light, \(3.0 \times 10^8 m/s\), the electron will be slower than photon at \(1.18 \times 10^8 m/s\) and the slowest will be the proton at \(7.73 \times 10^7 m/s\).
05

Calculate Momenta

The momentum of a particle is given by \(p = mv\), where \(m\) is the mass of the particle and \(v\) is its speed. For photons, the momentum is given by \(p = \frac{E}{c}\), where \(E\) is the energy of the photon and \(c\) is the speed of light. So the momenta for the photon, the electron and the proton are \(6.64 \times 10^{-34} kg \, m/s\), \(1.07 \times 10^{-22} kg \, m/s\), and \(1.29 \times 10^{-19} kg \, m/s\) respectively.
06

Compare Momenta

The proton has the greatest momentum (\(1.29 \times 10^{-19} kg \, m/s\)), while the photon has the least momentum (\(6.64 \times 10^{-34} kg \, m/s\)). Eventually, the solution for the exercise is derived.

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Most popular questions from this chapter

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