A 279 lb fullback runs the 40 yd dash at a speed of 18.5 ± 0.1 mi/h.(a) What is his de broglie wavelength (in meters)?(b)what is the uncertainty of his position?
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To calculate the de Broglie wavelength of the fullback, we can use the equation:
λ = h / p,
where λ is the de Broglie wavelength, h is the Planck constant (6.626 x 10^-34 J·s), and p is the momentum of the fullback.
To find the momentum (p), we need to convert the fullback's speed from miles per hour (mi/h) to meters per second (m/s). We can use the conversion factor 1 mi/h = 0.44704 m/s.
(a) Calculating the de Broglie wavelength:
1. Convert the speed from mi/h to m/s:
Speed = 18.5 ± 0.1 mi/h = (18.5 ± 0.1) x 0.44704 m/s
Speed = (18.5 x 0.44704) ± (0.1 x 0.44704) m/s
2. Calculate the momentum (p) using the formula:
Momentum (p) = mass x speed
The mass of the fullback is given as 279 lb, but we need to convert it to kg since the SI unit for mass is kilograms.
1 lb = 0.4536 kg
Mass = 279 lb x 0.4536 kg/lb
3. Calculate the momentum:
Momentum (p) = Mass x Speed
4. Calculate the de Broglie wavelength:
λ = h / p
(b) To find the uncertainty in position, we need to use the Heisenberg uncertainty principle, which states that the product of the uncertainty in position (Δx) and momentum (Δp) is greater than or equal to h/(4π).
The uncertainty in position (Δx) can be approximated as:
Δx ≈ h / (4πΔp),
where h is the Planck constant.
Now, let's calculate the answers step-by-step.
(a) Calculating the de Broglie wavelength:
1. Convert the speed from mi/h to m/s:
Speed = (18.5 x 0.44704) ± (0.1 x 0.44704) m/s
2. Convert the mass from lb to kg:
Mass = 279 lb x 0.4536 kg/lb
3. Calculate the momentum:
Momentum (p) = Mass x Speed
4. Calculate the de Broglie wavelength:
λ = h / p
(b) Calculating the uncertainty in position:
1. Calculate the uncertainty in momentum (Δp):
Δp = Mass x Uncertainty in speed
2. Calculate the uncertainty in position (Δx):
Δx ≈ h / (4πΔp)
Note: It's important to remember that the uncertainty in position calculated here is an approximation. In quantum mechanics, the Heisenberg uncertainty principle applies to the measurement uncertainty, not the uncertainty due to experimental uncertainties.