A baseball (1 kg) has an energy of 100 joules. If its uncertainty in position is 1 m, what is the percentage uncertainty (Δp/p × 100) in the momentum of the baseball?
A)

Question

A baseball (1 kg) has an energy of 100 joules. If its uncertainty in position is 1 m, what is the percentage uncertainty (Δp/p × 100) in the momentum of the baseball?
A) <<1%
B) 1%
C) 2%
D) 5%
E) 10%

Quizplus · Accepted Answer

The uncertainty in momentum (Δp) can be calculated using Heisenberg's uncertainty principle, ΔpΔx ≥ ħ/2, where Δx is the uncertainty in position and ħ is the reduced Planck's constant (approximately 1.05 × 10^-34 Js). Given Δx = 1 m, Δp ≥ (1.05 × 10^-34) / (2*1) = 5.25 × 10^-35 kg m/s. The momentum of the baseball (p) can be found from its kinetic energy (E = 100 J) using the relation E = p^2/(2m), where m is the mass of the baseball. Solving for p gives p = sqrt(2mE) = sqrt(2*1*100) = sqrt(200) kg m/s. The percentage uncertainty in momentum is then (Δp/p) × 100 = (5.25 × 10^-35 / sqrt(200)) × 100, which is much less than 1%, indicating that the uncertainty in the baseball's momentum due to its uncertainty in position is extremely small.

A Baseball (1 Kg) Has an Energy of 100 Joules