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 A = 100. pm?
To determine the speed of the baseball required to have a de Broglie wavelength equal to that of an x-ray photon, we can use the de Broglie wavelength equation:
λ = h / (mv)
where λ is the wavelength, h is the Planck's constant (6.626 x 10^(-34) J·s), m is the mass of the object, and v is the velocity of the object.
First, let's convert the atomic mass unit (amu) of the baseball to kilograms. The mass of the baseball is given as 138 g, which is equal to 0.138 kg (since 1 kg is equal to 1000 g).
Now, let's find the wavelength of the x-ray photon. The equation given for the wavelength is:
λ = A / 10^12
where λ is the wavelength in meters, and A is the wavelength in picometers (pm). We are given A = 100 pm, so substituting it into the equation:
λ = 100 pm / 10^12
Converting picometers to meters, 1 pm is equal to 1 x 10^(-12) m, so we have:
λ = (100 pm) / (10^12 pm/m)
Simplifying the expression:
λ = (100 / 10^12) m
Now we have the value for the wavelength of the x-ray photon.
To find the speed (velocity) of the baseball, we rearrange the de Broglie wavelength equation:
v = h / (mλ)
Substituting the given values:
v = (6.626 x 10^(-34) J·s) / ((0.138 kg) * (100 / 10^12 m))
v = (6.626 x 10^(-34) J·s) / ((0.138 kg) * (10^(-10) m))
v = (6.626 x 10^(-34) J·s) / (1.38 x 10^(-11) kg·m)
v ≈ 4.798 x 10^22 m/s
Therefore, the baseball must travel at approximately 4.798 x 10^22 m/s to have a de Broglie wavelength equal to that of an x-ray photon with A = 100 pm.