Question

A baseball with a mass of 150 g is moving at a velocity of 40 m/s (90 mph). If the uncertainty in the velocity is 0.1 m/s, the minimum uncertainty in position is ?

Answer 1

The uncertainty in position will be `4 * 10^(-29)"m"`.

As you know, Heisenberg's Uncertainty Principle states that one cannot simoultaneously measure with great precision both the momentum, and the position of a particle.

Mathematically, this is expressed as

`Deltap * Deltax>= h/(4pi)`, where

`Deltap` - the uncertainty in momentum;
`Deltax` - the uncertainty in position;
`h` - Planck's constant - `6.626 * 10^(-34)"m"^2"kg s"^(-1)`

The uncertainty in momentum can be written as

`Deltap = m * Deltav` `""`, where

`Deltav` - the uncertainty in velocity;
`m` - the mass of the particle.

In your case, you're dealing with a 150-g baseball that has an uncertainty in velocity of `"0.1 m/s"`, which means that the uncertainty in momentum will be

`Deltap = 150 * 10^(-3)"kg" * "0.1 m/s" = 15 * 10^(-3)"m kg s"^(-1)`

Now plug your data into the main equation and solve for `Deltax`

`Deltax >= h/(4pi) * 1/(Deltap)`

`Deltax >= (6.626 * 10^(-34)"m"^(cancel(2))cancel("kg")cancel("s"^(-1)))/(4 * pi * 15 * 10^(-3)cancel("kg")cancel("m")cancel("s"^(-1))) = 3.515 * 10^(-29)"m"`

Rounded to one sig fig, the number of sig figs given for the uncertainty in velocity, the answer will be

`Deltax >= color(green)(4 * 10^(-29)"m")`