If the uncertainties in the measurements of position and momentum are ...
Uncertainty principle:
The uncertainty principle states that it is impossible to simultaneously measure the exact position and momentum of a particle with complete accuracy. The more precisely we try to measure one of these quantities, the less precisely we can know the other.
Given information:
- The uncertainties in the measurements of position and momentum are equal.
- The mass of the particle is 1.21 x 10^-18 kg.
Mathematical representation:
The uncertainty principle is mathematically represented as Δx * Δp ≥ h/(4π), where Δx is the uncertainty in position, Δp is the uncertainty in momentum, and h is the Planck's constant (6.626 x 10^-34 Js).
Equal uncertainties:
Since the uncertainties in position and momentum are equal, we can write Δx = Δp.
Calculating the uncertainty in velocity:
Velocity is defined as the rate of change of position with respect to time. It is given by the equation v = Δx/Δt, where v is velocity, Δx is the change in position, and Δt is the change in time.
We can rearrange this equation to express the change in position as Δx = v * Δt.
Substituting the value of Δx in the uncertainty principle equation, we get Δp * v * Δt ≥ h/(4π).
Since Δp = Δx, we can rewrite the equation as Δx * v * Δt ≥ h/(4π).
Simplifying further, we have v * Δt ≥ h/(4πΔx).
The uncertainty in velocity is given by Δv = v * Δt.
Substituting the value of Δt in the equation, we get Δv = v * (h/(4πΔx))/v.
Simplifying further, we have Δv = h/(4πΔx).
Substituting Δx = Δp, we have Δv = h/(4πΔp).
Substituting the value of h and Δp, we get Δv = (6.626 x 10^-34 Js)/(4πΔp).
Since Δp = Δx, we can write Δv = (6.626 x 10^-34 Js)/(4πΔx).
Given that Δx = Δp, we can further simplify the equation to Δv = (6.626 x 10^-34 Js)/(4πΔp) = (6.626 x 10^-34 Js)/(4πΔx).
Calculating the uncertainty in velocity:
Substituting the value of Δx = Δp = Δv, we get Δv = (6.626 x 10^-34 Js)/(4πΔv).
Simplifying further, we have Δv^2 = (6.626 x 10^-34 Js)/(4π).
Taking the square root of both sides, we have Δv = √[(6.626 x 10^-34 Js)/(4π)].
Calculating the numerical value, we find Δv ≈ 6 m/s.
Therefore, the uncertainty in the measurement of velocity is approximately 6 m/s.
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