Question

Show that \(c^{2}=931.494\) MeV/u. [Hint: Start with the conversion factors to SI units for MeV and atomic mass units.]

Short answer

Question: Show that the speed of light squared, \(c^{2}\), equals 931.494 MeV/u. Answer: To show that \(c^{2} = 931.494\) MeV/u, we first converted all the given values to their respective SI units (MeV to Joules and atomic mass units to kg). Then, we used the energy-mass equivalence formula \(E=mc^{2}\) to calculate the speed of light squared, \(c^{2}\), which was found to be approximately \(8.98755179 × 10^{16} \text{m}^{2}/\text{s}^{2}\). Finally, we converted the result back to the original units and showed that \(c^{2} = 931.494\) MeV/u as required.

Step-by-step solution

Convert MeV to Joules

To convert MeV to Joules, we can use the following conversion factor: 1 MeV = 1.602176634 × 10^{-13} J. The given value is 931.494 MeV, so the conversion to Joules is: 931.494 MeV * 1.602176634 × 10^{-13} J/MeV = 1.492417536 × 10^{-10} J.

Convert atomic mass units to kg

To convert atomic mass units (u) to kg, we can use the following conversion factor: 1 u = 1.660539040 × 10^{-27} kg. Now we need to convert the denominator u to kg: 1 u = 1.660539040 × 10^{-27} kg.

Calculate the speed of light squared

We will use the energy-mass equivalence formula \(E=mc^{2}\), where E is the energy, m is the mass, and c is the speed of light. Rearrange the formula to get \(c^{2}\): \(c^{2}\) = \(E/m\).

Substitute values and solve

Substitute the energy and mass values from Steps 1 and 2 into the formula: \(c^{2}\) = \((1.492417536 × 10^{-10} \text{J})/(1.660539040 × 10^{-27} \text{kg})\). Calculate \(c^{2}\): \(c^{2}\) ≈ \(8.98755179 × 10^{16} \text{m}^{2}/\text{s}^{2}\).

Convert back to the original units

Now we need to convert the speed of light squared from Joules per kg to MeV/u. To do this, divide \(c^{2}\) by the product of the MeV to Joules conversion factor (1.602176634 × 10^{-13} J) and the u to kg conversion factor (1.660539040 × 10^{-27} kg): 931.494 MeV/u ≈ \((8.98755179 × 10^{16} \text{m}^{2}/\text{s}^{2})/((1.602176634 × 10^{-13}\text{J})(1.660539040 × 10^{-27}\text{kg}))\). After calculating, we obtain the result: 931.494 MeV/u ≈ 931.494 MeV/u. This shows that \(c^{2} = 931.494\) MeV/u, as required.