According to the de Broglie wavelength formula, the wavelength (λ) of a particle is inversely proportional to its momentum (p), where momentum is given by the product of mass (m) and velocity (v). Mathematically, the equation is:
λ = h / p,
where h is Planck's constant.
If we assume that the wavelengths of the proton and the electron are equal, we can set up the following relationship:
λ_proton = λ_electron.
Using the de Broglie equation, we can rewrite this as:
h / p_proton = h / p_electron.
Since the wavelengths are equal, we can equate the momenta:
p_proton = p_electron.
The momentum of a particle is given by the product of its mass and velocity:
p = m * v.
Now, we can compare the speeds of the proton (v_proton) and the electron (v_electron) using their respective momenta:
m_proton * v_proton = m_electron * v_electron.
Given that the mass of the proton (m_proton) is approximately 2000 times greater than the mass of the electron (m_electron):
2000 * m_electron * v_proton = m_electron * v_electron.
The mass of the electron cancels out, and we are left with:
2000 * v_proton = v_electron.
Therefore, if the wavelengths of the proton and electron are equal, the speed of the proton (v_proton) would be 2000 times smaller than the speed of the electron (v_electron). In other words, the electron would be moving approximately 2000 times faster than the proton under these conditions.