This problem involves the Heisenberg's uncertainty principle, which states that it is impossible to measure both position and velocity (or momentum) of a particle at the same time. The relationship between the uncertainties in the measurement is
`Deltax*Deltap > h/(4pi)`
Here, h is the Planck' constant: `h = 6.626*10^(-34) J*s` .
Please see the reference link to learn more about the uncertainty principle.
To find the uncertainty in the position of the electron, first we need to find the uncertainty in its momentum. Since the velocity is measured to an accuracy of 0.003%, the uncertainty of the velocity is
`0.003%*5*10^3 = 0.00003*5*10^3 = 0.15 m/s`
The uncertainty of the electron's momentum is then
`Delta p = m*0.15 = 9.1*10^(-31)*0.15 = 1.365*10^(-31)` kg*m/s.
Then, from the uncertainty principle, the uncertainty of the electron's position is at least
`Deltax=h/((4pi)*Deltap) = (6.626*10^(-34))/((4pi)*1.365*10^(-31))=3.86*10^(-4)` m.
The uncertainty in determining the position of electron is 0.386 millimeters.