We can solve the problem by using Heisenberg's uncertainty principle, which states that:
[tex]\Delta x \Delta p \geq \frac{h}{4 \pi} [/tex]
where
[tex]\Delta x[/tex] is the uncertainty on the position
[tex]\Delta p[/tex] is the uncertainty on the momentum
[tex]h[/tex] is the Planck constant.
Keeping in mind that the momentum is the product between the mass of the electron and its velocity:
[tex]p=mv[/tex]
we can rewrite the Heisenberg principle as
[tex]m \Delta x \Delta v \geq \frac{h}{4 \pi} [/tex]
where [tex]\Delta v[/tex] is the uncertainty on the velocity.
The uncertainty on the position is [tex]\Delta x = 587 pm = 587 \cdot 10^{-12} m[/tex], so we can find the uncertainty on the velocity by re-arranging the previous equation:
[tex]\Delta v \geq \frac{h}{4 \pi m \Delta x}= \frac{6.6 \cdot 10^{-34} Js}{4 \pi (9.1 \cdot 10^{-31}kg)(587 \cdot 10^{-12} m)} =9.84 \cdot 10^4 m/s[/tex]
