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Chapter 7: Problem 61

A particle has a velocity that is \(90 . \%\) of the speed of light. If the wavelength of the particle is \(1.5 \times 10^{-15} \mathrm{m},\) calculate the mass of the particle.

Short Answer

Expert verified
The mass of the particle is \(m_0 = \frac{\frac{6.626 \times 10^{-34} J \cdot s}{1.5 \times 10^{-15} m}}{0.9 \times 3 \times 10^8 m/s} \approx 1.63 \times 10^{-28} kg\).

Step by step solution

01

1. Identify the given information

The given information is the velocity of the particle (\(v = 0.9c\)) and the wavelength (\(λ = 1.5 \times 10^{-15}m\)).
02

2. Calculate the momentum using the de Broglie wavelength formula

Since \(λ = \frac{h}{p}\), we can solve for \(p\) (momentum) by rearranging the formula: \(p = \frac{h}{λ}\) Plug in the values \(h = 6.626 \times 10^{-34} J \cdot s\) (Planck's constant) and \(λ = 1.5 \times 10^{-15} m\): \(p = \frac{6.626 \times 10^{-34} J \cdot s}{1.5 \times 10^{-15} m}\)
03

3. Calculate the relativistic velocity

Now, we need to calculate the relativistic velocity using the relativistic length contraction formula \(λ_r = \frac{λ_0}{\sqrt{1-v^2/c^2}}\). First, rearrange the formula to solve for \(λ_0\): \(λ_0 = λ_r\sqrt{1-v^2/c^2}\) Since \(v = 0.9c\), we can make the following substitution: \(λ_0 = λ_r\sqrt{1-0.9^2}\) Plug in the given wavelength value, \(λ_r = 1.5 \times 10^{-15} m\): \(λ_0 = (1.5 \times 10^{-15} m)\sqrt{1-0.81}\)
04

4. Calculate the mass of the particle

Calculate the mass of the particle using the momentum and rest wavelength. The momentum formula is given by \(p = m_0v\), where \(m_0\) is the rest mass of the particle. Rearrange the formula to solve for \(m_0\): \(m_0 = \frac{p}{v}\) Plug in the values for momentum from step 2 and velocity from step 3: \(m_0 = \frac{\frac{6.626 \times 10^{-34} J \cdot s}{1.5 \times 10^{-15} m}}{0.9 \times 3 \times 10^8 m/s}\)
05

5. Calculate the rest mass of the particle

Now we can calculate the rest mass by plugging in the values obtained in previous steps into the following equation: \(m_0 = \frac{\frac{6.626 \times 10^{-34} J \cdot s}{1.5 \times 10^{-15} m}}{0.9 \times 3 \times 10^8 m/s}\) Calculate the mass to obtain the final answer.

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