Calculate the de Broglie wavelength, in nanometers, associated with a 145 {\rm g} baseball traveling at a speed of 162km/h
wavelength = h/mv
To calculate the de Broglie wavelength, we will use the de Broglie equation:
λ = h / p
where λ represents the wavelength, h is the Planck constant (6.626 x 10^-34 J⋅s), and p is the momentum of the baseball.
To find the momentum (p), we need to calculate the velocity (v) of the baseball. Given that the speed of the baseball is 162 km/h, we need to convert it to meters per second (m/s) since SI units are used in the de Broglie equation.
1 km = 1000 m
1 h = 3600 s
So, the velocity of the baseball in m/s would be:
v = 162 km/h * (1000 m/km) / (3600 s/h) = 45 m/s
Next, we can find the momentum (p) of the baseball using the formula:
p = m * v
where m is the mass of the baseball and v is the velocity.
Given that the mass of the baseball is 145 g, we need to convert it to kilograms (kg) since SI units are used in the de Broglie equation.
1 g = 0.001 kg
So, the mass of the baseball in kg would be:
m = 145 g * (0.001 kg/g) = 0.145 kg
Now we can calculate the momentum:
p = 0.145 kg * 45 m/s = 6.525 kg⋅m/s
Finally, we can substitute the values of h and p into the de Broglie equation to find the wavelength (λ) of the baseball:
λ = 6.626 x 10^-34 J⋅s / 6.525 kg⋅m/s = 1.014 x 10^-34 m
To convert the wavelength to nanometers, we multiply by 10^9:
λ = 1.014 x 10^-34 m * (10^9 nm/m) = 1.014 x 10^-25 nm
Therefore, the de Broglie wavelength of the baseball is approximately 1.014 x 10^-25 nm.