Final answer:
To find the uncertainty in the position of a proton with a known uncertainty in velocity, we apply the Heisenberg Uncertainty Principle. By calculating the uncertainty in momentum as the product of the proton's mass and the velocity uncertainty, we can solve for the position uncertainty. This yields the minimum possible uncertainty in the proton's position in meters.
Step-by-step explanation:
To determine the uncertainty in the position of a proton when its velocity uncertainty is given, we can use the Heisenberg Uncertainty Principle, which relates the uncertainties in position (Δx) and momentum (Δp). The principle can be expressed as Δx * Δp ≥ ħ/2, where ħ (h-bar) is the reduced Planck constant (ħ = h / (2π) = 1.055 × 10-34 kg·m²/s). Since momentum p is mass m times velocity v, the uncertainty in momentum Δp can be written as m * Δv. For a proton (mass approximately 1.673 × 10-27 kg), given an uncertainty in velocity Δv of 7.40 × 10-4 m/s, the uncertainty in momentum Δp is (1.673 × 10-27 kg) * (7.40 × 10-4 m/s). Subsequently, we can rearrange the Heisenberg uncertainty equation to solve for the uncertainty in position: Δx ≥ ħ/(2 * Δp). Plugging in the values, we calculate Δx as approximately 1.055 × 10-34 kg·m²/s divided by (2 * (1.673 × 10-27 kg) * (7.40 × 10-4 m/s)), which gives us the minimum possible uncertainty in the proton's position in meters.