Question

A proton has a momentum of \(3.0 \mathrm{GeV} / c\). With what velocity is it moving relative to a stationary observer? a) \(0.31 c\) c) \(0.91 c\) e) \(3.2 c\) b) \(0.33 c\) d) \(0.95 c\)

Short answer

Solution: After converting the momentum to kg·m/s and using the relativistic momentum formula, we calculate the velocity of the proton to be approximately 0.955c. Comparing this result to the given options, the closest option is (d) 0.95c. Therefore, the correct answer is: d) 0.95c

Step-by-step solution

Identify known variables and the desired value

We know the following: - The momentum of the proton: \(3.0 \: \mathrm{GeV/c}\) - The mass of the proton is about \(1.67 \times 10^{-27} \: \mathrm{kg}\) - The speed of light is \(3.0 \times 10^8 \: \mathrm{m/s}\) We want to find the velocity of the proton relative to the stationary observer.

Convert the momentum value

First, let's convert the given momentum from \(\mathrm{GeV/c}\) to \(\mathrm{kg\:m/s}\). We know that 1 GeV = \(1\times10^9\) eV, and 1 eV = \(1.6\times10^{-19}\) J, we can make the conversion. Therefore, the momentum of the proton equals: \(p = 3.0\:\mathrm{GeV/c} \cdot \frac{1\times10^9 \: \mathrm{eV}}{\mathrm{GeV}} \cdot \frac{1.6\times10^{-19} \: \mathrm{J}}{\mathrm{eV}} \cdot \frac{1}{c}\)

Use the relativistic momentum formula

The momentum of a relativistic particle can be written as: \(p = \gamma m_0v\) where \(m_0\) is the rest mass of the particle, \(v\) is the velocity of the particle, and \(\gamma\) is the Lorentz factor given by: \(\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}\) Our goal is to find the velocity \(v\), so we need to solve the above equation for \(v\).

Rearrange the formula to find \(v\)

First, we can find \(\gamma m_0v\). Then, we can rearrange the momentum formula to solve for \(v\): \(\gamma m_0v = \frac{p}{m_0} = \frac{3.0\:\mathrm{GeV/c} \cdot \frac{1\times10^9 \: \mathrm{eV}}{\mathrm{GeV}} \cdot \frac{1.6\times10^{-19} \: \mathrm{J}}{\mathrm{eV}} \cdot \frac{1}{c}}{1.67 \times 10^{-27} \: \mathrm{kg}}\) Solve for \(\gamma v\): \(\gamma v = \frac{p}{m_0c}\) Now we need to substitute the Lorentz factor formula for \(\gamma\) to get: \(v = \frac{p}{m_0c\sqrt{1 - \frac{v^2}{c^2}}}\)

Solve the equation for \(v\) numerically

To find \(v\), we need to solve the equation numerically. We can use an iterative method or a numerical solver to find the solution. Upon solving the equation, we obtain: \(v \approx 0.955 c\)

Compare the calculated velocity with the given options and select the correct answer

The calculated velocity is close to the option (d). Therefore, the correct answer is: d) \(0.95 c\)