To what uncertainty (in m) can the position of a baseball traveling at 45.0 m/s be measured if the uncertainty of its speed is 0.10%? The mass of a baseball is about 0.145 kg.

Δx ≥ 8 x 10^(-29) m

Explanation:

We will solve this using Heisenberg's Uncertainty Principle which states that one cannot simultaneously measure with great precision both the momentum, and the position of a particle.

Thus, mathematically, this is expressed as

Δp ⋅ Δx ≥ h/4π

where;

Δp is the uncertainty in momentum;

Δx is the uncertainty in position;

h is Planck's constant which has a value of 6.626 x 10^(−34) m²kg/s

Furthermore, the uncertainty in momentum can be written as;

Δp = m ⋅ Δv

where Δv is the uncertainty in velocity while m is the mass of the particle.

In this question, the mass of the baseball is 0.145kg or 145g with an uncertainty in velocity of 0.1%

So, uncertainty in velocity = 0.1% x 45 = 0.045 m/s

Thus, the uncertainty in momentum will be;

Δp = 0.145kg x 0.045 m/s = 6.525 x 10^(-3) m.kg/s

Now, let's plug in the relevant data into the Uncertainty Principle equation and make Δx the subject.

Thus;

Δx ≥ (h/4π) x (1/Δp)

Δx ≥ [(6.626 x 10^(−34))/(4π)] /(1/(6.525 x 10^(-3)))

= 8.08 x 10^(-29) m

If we round to one sig fig, the uncertainty in velocity, will be

Δx ≥ 8 x 10^(-29) m

3. dark matter

empty space could also be a simple vacuum