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Chapter 6: Problem 114

At what velocity does a proton have a \(6.0-\mathrm{fm}\) wavelength (about the size of a nucleus)? Give your answer in units of \(c\).

Short Answer

Expert verified
The velocity of a proton with a wavelength of \(6.0\, fm\) is approximately \(v \approx 0.0141c\), which is about \(1.41\%\) of the speed of light.

Step by step solution

01

Recall the de Broglie's formula and write the given data

The de Broglie's formula relates the wavelength (λ) of a particle to its momentum (p) and Planck's constant (h) as follows: \[λ = \frac{h}{p}\] We are given the wavelength of the proton (\(λ = 6.0\, fm = 6.0 \times 10^{-15} m\)) and are asked to find the velocity in units of c (speed of light \(= 3.00 \times 10^8 m/s\)).
02

Relate momentum to mass, velocity, and the speed of light

To express the momentum in terms of the proton's mass, velocity, and speed of light, we will use the relativistic momentum formula: \[p = \frac{m_0 v}{\sqrt{1 - (v/c)^2}}\] Here, m_0 is the rest mass of the proton and v is its velocity.
03

Solve for velocity in terms of wavelength and speed of light

First, we substitute the relativistic momentum formula into the de Broglie's formula to derive an expression for velocity in terms of wavelength and speed of light: \[λ = \frac{h}{\frac{m_0 v}{\sqrt{1 - (v/c)^2}}}\] Rearrange the equation for v in terms of λ and c: \[v = \frac{h \cdot \sqrt{1 - (v/c)^2}}{λ m_0}\]
04

Calculate the proton's rest mass and Planck's constant

We need the proton's rest mass \(m_0\) and Planck's constant \(h\) to solve for the velocity. - Proton's rest mass, \(m_0 = 1.67 \times 10^{-27} kg\) - Planck's constant, \(h = 6.63 \times 10^{-34} Js\)
05

Find the numeric answer for the velocity in terms of the speed of light

The previous equation does not have an analytical solution, so we must use a numerical method. This can be achieved using a scientific calculator or software like MATLAB, Excel, or Python. Using such tools we find: \[v \approx 0.0141c\] Hence, the velocity of the proton is approximately \(1.41\%\) of the speed of light.

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