How fast must a 138 g baseball travel in order to have a de Broglie wavelength that is equal to that of an x-ray photon with = 100. pm?
To determine the speed at which a 138 g baseball must travel in order to have a de Broglie wavelength equal to that of an x-ray photon with a wavelength of 100. pm, we can use the de Broglie wavelength equation:
λ = h / p
Where λ is the wavelength, h is the Planck's constant (6.626 x 10^-34 J·s), and p is the momentum.
First, let's convert the given mass of the baseball from grams to kilograms:
Mass of baseball = 138 g = 0.138 kg
Now, we need to find the momentum of the x-ray photon. The momentum of a photon can be calculated as:
p = E / c
Where E is the energy and c is the speed of light.
Given that the wavelength of the x-ray photon is 100. pm (picometers), we can use the formula:
λ = c / ν
Where λ is the wavelength, c is the speed of light, and ν is the frequency.
Since we know the wavelength of the x-ray photon, we can find the frequency using the formula:
ν = c / λ
Substituting the given values:
ν = (3.0 x 10^8 m/s) / (100 x 10^-12 m) = 3.0 x 10^16 Hz
Now we have the frequency of the x-ray photon. To calculate its energy, we can use the equation:
E = hν
Substituting the values:
E = (6.626 x 10^-34 J·s)(3.0 x 10^16 Hz) = 1.9878 x 10^-17 J
Next, we can find the momentum of the x-ray photon:
p = E / c = (1.9878 x 10^-17 J) / (3.0 x 10^8 m/s) ≈ 6.626 x 10^-26 kg·m/s
Now, we can calculate the de Broglie wavelength of the baseball using the given mass:
λ = h / p = (6.626 x 10^-34 J·s) / (0.138 kg)(v)
Rearranging the equation, we can solve for v:
v = h / (0.138 kg)(λ)
Substituting the values:
v = (6.626 x 10^-34 J·s) / (0.138 kg)(100 x 10^-12 m) ≈ 4.82 x 10^-6 m/s
So, the baseball must travel at approximately 4.82 x 10^-6 m/s in order to have a de Broglie wavelength equal to that of an x-ray photon with a wavelength of 100. pm.