Question

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

(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

Calculate the Speed of the Photon

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

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 \(v = \sqrt{\frac{2 \times 6.4\times10^{-14} J}{9.1\times10^{-31} kg}} = 1.18 \times 10^8 \, m/s\).

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 \(v = \sqrt{\frac{2 \times 1.6 \times 10^{-12} J}{1.67 \times 10^{-27} kg}} = 7.73 \times 10^7 \, m/s\).

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 \(7.73 \times 10^7 m/s\).

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.

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.

Key concepts

Photon Properties

Photons are fascinating particles that are fundamental to our understanding of light and electromagnetic radiation. Unlike most particles, photons have a unique property – they possess no mass at all. This remarkable trait makes photons extremely interesting when studying particle motion and properties. Although they are massless, they do carry energy and momentum.
A photon moves through space at an incredible speed. In fact, photons always travel at the speed of light, which is approximately \(3.0 \times 10^8 \, \text{m/s}\). This speed is constant and does not change regardless of how much energy the photon possesses. Furthermore, because photons do not have mass, their momentum is calculated differently compared to other particles. The formula for momentum is \(p = \frac{E}{c}\), where \(E\) represents the energy of the photon and \(c\) stands for the speed of light. This shows that a photon's momentum is directly dependent on its energy, unlike typical particles where mass and velocity determine momentum.

Electron and Proton Speeds

When comparing the speeds of an electron and a proton, it's crucial to take into account their energy and mass. Electrons are subatomic particles with a very small mass of approximately \(9.1 \times 10^{-31} \, \text{kg}\). In the case of our example, the electron has an energy of \(0.40 \, \text{MeV}\). By converting this energy to joules, we can then calculate its speed using the kinetic energy formula: \(v = \sqrt{\frac{2E}{m}}\). With this calculation, we find that the electron moves at a speed of \(1.18 \times 10^8 \, \text{m/s}\).
On the other hand, protons are much more massive than electrons, with a mass of about \(1.67 \times 10^{-27} \, \text{kg}\). Given a kinetic energy of \(10 \, \text{MeV}\), we can similarly compute the speed of the proton. Following the same process yields a speed of \(7.73 \times 10^7 \, \text{m/s}\). Compared to the electron and photon, the proton is significantly slower due to its larger mass, requiring more energy to achieve the same speed.

Momentum Comparison

Momentum is a key concept when studying particle motion, providing a deeper understanding of how mass and velocity influence movement. The momentum of a particle is typically calculated using the equation \(p = mv\). However, this changes for massless particles like photons, where we use the alternative formula \(p = \frac{E}{c}\).
In the exercise, when we calculate the momenta for each of the following: a photon (at \(6.64 \times 10^{-34} \, \text{kg} \, \text{m/s}\)), an electron (\(1.07 \times 10^{-22} \, \text{kg} \, \text{m/s}\)), and a proton (\(1.29 \times 10^{-19} \, \text{kg} \, \text{m/s}\)), an interesting comparison emerges. Despite the proton moving slowest in terms of speed, it has the greatest momentum due to its larger mass.

• The photon, being massless, has the smallest momentum because its energy is comparably low.
• Although the electron has a smaller mass than the proton, its higher speed allows its momentum to be greater than that of the photon yet significantly less than the proton's.
• This comparison illustrates how mass and velocity collectively determine a particle's momentum.

Understanding these differences in momentum can offer valuable insights in fields such as particle physics and quantum mechanics.