A 295 lb fullback runs the 40 yd dash at a speed of 19.3 ± 0.1 mi/h.
(a) What is his de Broglie wavelength (in meters)?
(b) What is the uncertainty in his position?
wavelength = h/mv
You will need to convert 295 lbs to kg and convert 19.3 mi/hr into m/s.
ok what do you do for part b)
To calculate the de Broglie wavelength of the fullback in meters, we can use the following formula:
λ = h / p
where λ is the wavelength, h is Planck's constant (6.626 x 10^-34 J·s), and p is the momentum of the fullback.
To find the momentum of the fullback, we can use the equation:
p = m · v
where m is the mass of the fullback and v is the velocity.
(a) Calculating the de Broglie wavelength:
First, we need to convert the velocity from miles per hour (mi/h) to meters per second (m/s).
1 mi/h = 0.44704 m/s
So, the speed of the fullback in meters per second is:
v = 19.3 mi/h * 0.44704 m/s / 1 mi/h ≈ 8.63 m/s
Next, we calculate the momentum:
p = m · v = 295 lb * 0.45359 kg/lb * 8.63 m/s ≈ 1151.97 kg·m/s
Finally, we can calculate the de Broglie wavelength:
λ = h / p ≈ (6.626 x 10^-34 J·s) / (1151.97 kg·m/s) ≈ 5.75 x 10^-38 m
Therefore, the de Broglie wavelength of the fullback is approximately 5.75 x 10^-38 meters.
(b) Calculating the uncertainty in his position:
The uncertainty in position can be estimated using the Heisenberg uncertainty principle:
Δx Δp ≥ h / (4π)
where Δx is the uncertainty in position, Δp is the uncertainty in momentum, and h is Planck's constant.
The uncertainty in momentum can be determined using the uncertainty in velocity:
Δp = m · Δv
To estimate Δv, we can use the uncertainty in speed provided, which is ± 0.1 mi/h.
Converting this uncertainty to meters per second:
Δv = ± 0.1 mi/h * 0.44704 m/s / 1 mi/h ≈ ± 0.045 m/s
Now, we calculate the uncertainty in momentum:
Δp = m · Δv = 295 lb * 0.45359 kg/lb * 0.045 m/s ≈ 5.06 kg·m/s
Finally, using the Heisenberg uncertainty principle:
Δx Δp ≥ h / (4π)
Δx * 5.06 kg·m/s ≥ (6.626 x 10^-34 J·s) / (4π)
Δx * 5.06 kg·m/s ≥ 5.27 x 10^-35 m
Δx ≥ (5.27 x 10^-35 m) / (5.06 kg·m/s)
Δx ≥ 1.04 x 10^-35 m
Therefore, the uncertainty in the fullback's position is approximately 1.04 x 10^-35 meters.
To calculate the de Broglie wavelength and uncertainty in position, we need to use the principles of quantum mechanics. The de Broglie wavelength is given by the equation:
λ = h / p
where λ represents the de Broglie wavelength, h is the Planck's constant (6.626 x 10^-34 J s), and p is the momentum of the object.
(a) Calculating the de Broglie wavelength:
To find the momentum (p) of the fullback, we can use the formula:
p = m * v
where p is momentum, m is mass, and v is velocity.
First, we need to convert the fullback's weight from pounds (lb) to kilograms (kg). The conversion factor is 1 lb = 0.4536 kg.
So, the fullback's mass (m) is:
m = 295 lb * 0.4536 kg/lb
Next, we need to convert the speed from miles per hour (mi/h) to meters per second (m/s). The conversion factor is 1 mi/h = 0.44704 m/s.
So, the speed (v) of the fullback is:
v = 19.3 mi/h * 0.44704 m/s
Now, we can calculate the momentum (p):
p = m * v
Finally, we can substitute the calculated momentum into the de Broglie wavelength equation to find λ.
(b) The uncertainty in position is related to the uncertainty in momentum. According to the Heisenberg uncertainty principle, the product of the uncertainty in position (Δx) and the uncertainty in momentum (Δp) must be greater than or equal to h/4π.
Δx * Δp ≥ h / (4π)
Solving for Δx, we can find the uncertainty in position.
Now let's calculate both the de Broglie wavelength and the uncertainty in position:
(a) Calculating the de Broglie wavelength:
- Convert the fullback's weight from pounds (lb) to kilograms (kg):
m = 295 lb * 0.4536 kg/lb
- Convert the speed from miles per hour (mi/h) to meters per second (m/s):
v = 19.3 mi/h * 0.44704 m/s
- Calculate the momentum using the formula:
p = m * v
- Plug in the values of h and p in the de Broglie wavelength equation:
λ = h / p
(b) Calculating the uncertainty in position:
- Use the Heisenberg uncertainty principle equation:
Δx * Δp ≥ h / (4π)
- Solve for Δx to find the uncertainty in position.