# A measurement of an electron's speed is `v = 2.0 x 10^6 m/s` and has an uncertainty of `10%.` What is the minimum uncertainty in its position? (`h = 6.626 x 10^-34 J*s,`  `m_el = 9.11 x 10^-31 kg`). Show steps. I got `.91 nm,` is it correct?

Borys Shumyatskiy | Certified Educator

calendarEducator since 2015

starTop subjects are Math and Science

Hello!

The uncertainty principle (one of them) states that it is impossible to exactly measure the speed and the position of a body. This inexactness is called uncertainty. Of course it is significant only for very small particles of matter, for example for electrons.

The main formula for the speed-position uncertainty is

`Delta p*Delta x gt= bar h/2,`

where `Delta p` is the uncertainty of the momentum, `Delta x` is the uncertainty of the position and `bar h` is the reduced Plank's constant  `h/(2pi).` ` `

The momentum is the mass multiplied by the speed and `10%` is `0.1.` Thus we obtain the inequality

`0.1*m_(el)*v*Delta x gt= h/(4pi),`

and therefore

`Delta x gt= h/(4pi)*10/(m_(el)*v) =(6.626*10^(-34))/(4pi)*10/(9.11*10^(-31)*2*10^6)=`

`=(6.626)/(4pi*9.11*2)*10^(-8) approx0.0289*10^(-8) = 2.89*10^(-10) (m).`

In nanometers it is `0.289 nm.`