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.
To find the minimum uncertainty in the momentum of an object, we can 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 Planck's constant divided by 4π.
(a) To find the minimum uncertainty in the momentum of the object, we can rearrange the uncertainty principle equation as follows:
Δx * Δp ≥ h / 4π
Given that the uncertainty in position (Δx) is 2.40 m, we can plug in the values and solve for Δp:
2.40 m * Δp ≥ h / 4π
Δp ≥ h / (4π * 2.40 m)
Now, we can substitute the value of Planck's constant (h = 6.63 × 10^-34 Js) into the equation:
Δp ≥ (6.63 × 10^-34 Js) / (4π * 2.40 m)
Simplifying the expression:
Δp ≥ 2.95 × 10^-35 kg·m/s
Therefore, the minimum uncertainty in the momentum of the object is 2.95 × 10^-35 kg·m/s.
(b) To find the minimum uncertainty in the object's velocity assuming it is a golf ball, we can use the definition of momentum (p = mv), where m is the mass of the object and v is its velocity.
Since we already have the minimum uncertainty in momentum (Δp), we can use it to find the minimum uncertainty in velocity (Δv). The uncertainty in velocity can be obtained by dividing the uncertainty in momentum by the mass of the object.
Given that the mass of a golf ball is 0.0450 kg, we can calculate the minimum uncertainty in velocity as follows:
Δv = Δp / m
Δv = (2.95 × 10^-35 kg·m/s) / 0.0450 kg
Δv ≈ 6.56 × 10^-34 m/s
Therefore, the minimum uncertainty in the golf ball's velocity is approximately 6.56 × 10^-34 m/s.
(c) Similarly, to find the minimum uncertainty in the object's velocity assuming it is an electron, we can again use the formula Δv = Δp / m.
The mass of an electron is 9.11 × 10^-31 kg. We already know the minimum uncertainty in momentum (Δp) from part (a), which is 2.95 × 10^-35 kg·m/s. Plugging in these values, we can calculate the minimum uncertainty in velocity:
Δv = Δp / m
Δv = (2.95 × 10^-35 kg·m/s) / (9.11 × 10^-31 kg)
Δv ≈ 3.24 × 10^-4 m/s
Therefore, the minimum uncertainty in the electron's velocity is approximately 3.24 × 10^-4 m/s.