Final answer:
The uncertainty in position of the dust particle is given as 1 × 10^-12 m. Using the uncertainty principle and the equations for uncertainty in momentum and velocity, the uncertainty in momentum is approximately 5.27 × 10^-18 kg m/s and the uncertainty in velocity is approximately 5.27 × 10^-12 m/s.
Step-by-step explanation:
The uncertainty principle, established by Werner Heisenberg, states that there is a limit to how precisely we can know both the position and momentum of a particle.
The uncertainty in position, denoted as Δx or Ax, is equal to the accuracy of the measurement.
In this case, the uncertainty in position of the dust particle is given as Δx = 1 pm = 1 × 10-12 m.
The uncertainty in momentum, denoted as Δp or Ap, can be calculated using the relation ΔxΔp ≥ h/4π, where h is the Planck's constant.
Rearranging the equation to solve for Δp, we have Δp = h/(4πΔx).
Once we have the uncertainty in momentum, we can find the uncertainty in velocity using the relation Δp = mΔv, where m is the mass of the dust particle.
Rearranging the equation to solve for Δv, we have Δv = Δp/m.
Plugging in the values, we have
Δp = (6.63 × 10-34 kg m2/s)/(4π × 1 × 10-12 m) and
Δv = Δp/(1 × 10-6 kg),
which gives us:
Δp ≈ 5.27 × 10-18 kg m/s and Δv ≈ 5.27 × 10-12 m/s