An object is moving along a straight line, and the uncertainty in its position is 2.40 m. (a) Find the minimum uncertainty in the momentum of the object. Find the minimum uncertainty in the object's velocity, assuming that the object is (b) a golf ball (mass = 0.0450 kg) and (c) an electron.
(a) The minimum uncertainty in the momentum p can be found by employing Heisenberg's Uncertainty Principle:
(Δx)(Δp) = ħ/2
We are given the uncertainty in position Δx and want to find the uncertainty in momentum Δp. Rearranging the equation to solve for Δp gives:
Δp = ħ/(2Δx)
Where ħ is the reduced Planck constant, ħ = 1.054571818x10^(-34) J s. We are given Δx = 2.40 m. Plugging it into the equation:
Δp = 1.054571818x10^(-34) J s / (2 * 2.40 m)
Δp = 2.19411412x10^(-35) kg m/s
The uncertainty in the object's momentum is 2.19x10^(-35) kg m/s.
(b) To find the minimum uncertainty in the velocity of a golf ball (mass = 0.0450 kg), we use the formula:
Δv = Δp/m
Where Δp = 2.19x10^(-35) kg m/s is the uncertainty in the momentum and m = 0.0450 kg is the mass of the golf ball.
Δv = 2.19x10^(-35) kg m/s / 0.0450 kg
Δv = 4.87x10^(-34) m/s
The uncertainty in the golf ball's velocity is 4.87x10^(-34) m/s.
(c) To find the minimum uncertainty in the velocity of an electron, we use the same formula for Δv:
Δv = Δp/m
Now, the mass of the electron, m = 9.11x10^(-31) kg. Plugging that into the equation:
Δv = 2.19x10^(-35) kg m/s / 9.11x10^(-31) kg
Δv = 2.40x10^(4) m/s
The uncertainty in the electron's velocity is 2.40x10^(4) m/s.
(a) The uncertainty principle states that ΔxΔp ≥ ħ/2, where Δx is the uncertainty in position, Δp is the uncertainty in momentum, and ħ is the reduced Planck's constant (ħ = h/2π).
Given that Δx = 2.40 m, we can calculate the minimum uncertainty in momentum:
ΔxΔp ≥ ħ/2
2.40 m Δp ≥ ħ/2
Δp ≥ ħ/(2.4 m)
Now, let's substitute the value of ħ:
Δp ≥ (h/2π)/(2.4 m)
Δp ≥ h/(2π * 2.4 m)
(b) Let's calculate the minimum uncertainty in momentum for a golf ball with mass m = 0.0450 kg.
Δp ≥ h/(2π * 2.4 m)
Δp ≥ (6.63 × 10^(-34) J·s) / (2π * 2.4 m)
Δp ≥ 1.398 × 10^(-34) J·s / m
Now, we know that momentum (p) is given by p = mv, where m is the mass and v is the velocity. Therefore, the minimum uncertainty in velocity (Δv) can be found using the relation Δv = Δp / m:
Δv = Δp / m
Δv ≥ (1.398 × 10^(-34) J·s / m) / (0.0450 kg)
(c) Now, let's calculate the minimum uncertainty in momentum and velocity for an electron with a mass of m = 9.11 × 10^(-31) kg.
Δp ≥ h/(2π * 2.4 m)
Δp ≥ (6.63 × 10^(-34) J·s) / (2π * 2.4 m)
Δp ≥ 1.398 × 10^(-34) J·s / m
Δv = Δp / m
Δv ≥ (1.398 × 10^(-34) J·s / m) / (9.11 × 10^(-31) kg)
Now we have the minimum uncertainties in momentum and velocity for both the golf ball and the electron.
To find the minimum uncertainty in the momentum of an object, you need to use the uncertainty principle, which states that the product of the uncertainty in position and the uncertainty in momentum must be greater than or equal to the reduced Planck's constant, h/2π.
(a) To find the minimum uncertainty in the momentum of the object, you need to determine the uncertainty in position first. The uncertainty in position is given to be 2.40 m.
According to the uncertainty principle, h/2π = Δx * Δp, where Δx is the uncertainty in position and Δp is the uncertainty in momentum.
We need to rearrange this formula to solve for Δp:
Δp = (h/2π) / Δx.
Substituting the given values, we get:
Δp = (6.626 x 10^-34 J·s / (2 x 3.14159)) / 2.40 m.
Calculating this expression gives us the minimum uncertainty in momentum for the object.
(b) To find the minimum uncertainty in the object's velocity when it is a golf ball (mass = 0.0450 kg), we need to recall that momentum (p) is given by the equation p = mv (where m is mass and v is velocity).
Therefore, the minimum uncertainty in velocity can be found by dividing the minimum uncertainty in momentum (calculated in step a) by the mass of the golf ball:
Δv = Δp / m.
(c) To find the minimum uncertainty in the object's velocity when it is an electron, we need to use the mass of an electron, which is 9.11 x 10^-31 kg.
Again, we divide the minimum uncertainty in momentum (calculated in step a) by the mass of the electron:
Δv = Δp / m.
By performing these calculations, we can find the minimum uncertainty in velocity for both the golf ball and the electron.