An air rifle is used to shoot 1.0 g particles at 100 m/s through a hole of diameter 2.0 mm. How far
from the rifle must an observer be to see the beam spread by 1.0 cm because of the uncertainty
principle? Compare this answer with the diameter of the universe ( 1026 m).
To calculate the distance from the rifle where an observer can see the beam spread by 1.0 cm due to the uncertainty principle, we can use the following steps:
Step 1: Calculate the wavelength of the particle.
The wavelength (λ) of a particle can be calculated using the de Broglie wavelength equation:
λ = h / p
where h is the Planck's constant (6.626 x 10^-34 J·s) and p is the momentum of the particle. The momentum (p) can be calculated using the formula:
p = m * v,
where m is the mass of the particle (1.0 g) and v is the velocity of the particle (100 m/s).
mass (m) = 1.0 g = 1.0 x 10^-3 kg
velocity (v) = 100 m/s
p = m * v
p = (1.0 x 10^-3) * (100)
p = 0.1 kg·m/s
λ = h / p
λ = (6.626 x 10^-34) / (0.1)
λ ≈ 6.626 x 10^-33 m
Step 2: Calculate the angle of dispersion.
The angle of dispersion (θ) can be calculated using the formula:
θ = λ / D,
where D is the diameter of the hole (2.0 mm = 2.0 x 10^-3 m).
D = 2.0 x 10^-3 m
θ = λ / D
θ = (6.626 x 10^-33) / (2.0 x 10^-3)
θ ≈ 3.313 x 10^-30 radians
Step 3: Calculate the distance from the observer.
To calculate the distance from the observer (d) at which the beam spread is 1.0 cm, we can consider the small angle approximation:
d ≈ x / θ,
where x is the beam spread (1.0 cm = 1.0 x 10^-2 m).
x = 1.0 x 10^-2 m
d ≈ x / θ
d ≈ (1.0 x 10^-2) / (3.313 x 10^-30)
d ≈ 3.018 x 10^27 m
The distance from the rifle where an observer can see the beam spread by 1.0 cm is approximately 3.018 x 10^27 m.
Comparing this answer with the diameter of the universe (approximately 10^26 m), we can see that the distance is much greater than the diameter of the universe.
To calculate the distance from the rifle at which an observer can see the beam spread by 1.0 cm due to the uncertainty principle, you need to consider the de Broglie wavelength of the particles and the angle of spread.
The de Broglie wavelength is given by the 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 of the particle.
The momentum (p) of the particle can be calculated using the equation:
p = m * v
where m is the mass of the particle (1.0 g = 0.001 kg) and v is the velocity of the particle (100 m/s).
Now, to find the angle of spread, you can use the equation for angular spread:
θ = λ / d
where θ is the angle of spread, λ is the wavelength, and d is the diameter of the hole.
Substituting the values, we have:
θ = (h / (m * v)) / d
Next, we need to calculate the distance (x) at which the beam spreads by 1.0 cm. We can use the following trigonometric relationship:
tan(θ) = opposite / adjacent
In this case, the opposite is 1.0 cm (0.01 m) and the adjacent is the distance from the rifle (x). Rearranging the equation, we have:
x = opposite / tan(θ)
Substituting the values, we get:
x = 0.01 m / tan(θ)
Finally, we can compare this distance with the diameter of the universe (10^26 m) to see the scale of the result.
Please note that this calculation assumes that the particles are non-interacting and have no forces acting on them other than their initial momentum.