## Baseball ball

Baseball ball with the mass (150 ± 1) g was thrown with the velocity (40 ± 1) m·s−1.

What is the smallest inevitable uncertainty when determining its location?

For simplicity assume just one-dimensional motion.

• #### Hint 1

We know that, using the principle of uncertainty, with the specification of a particle’s position comes the price of lost accuracy in measuring another quantity. What is the quantity (or quantities) in question?

• #### Hint 2

What is the relative uncertainty of a measurement? How does the relative uncertainty propagate from one quantity to another? What if the one quantity is the product of two other which already carry some uncertainty?

• #### Notation

 m = 150 g Ball’s mass Δm = 1 g Absolute uncertainty of the mass v = 40 m·s−1 Ball’s velocity Δv = 1 m·s−1 Absolute uncertainty of the velocity Δx = ? (m) Uncertainty of the location
• #### Solution

We need to know the uncertainty Δp to estimate Δx from the principle of uncertainty $\Delta x \,\cdot\, \Delta p \ge \frac{\hbar}{2}$ . As uncertainty of the momentum, we consider the absolute uncertainty of the determination of the momentum. Since the ball’s momentum is the product of its mass and velocity  p = m·v we can use this formula to calculate its relative uncertainty of these variables η(p) = η(m) + η(v).
We have also formula from the definition of relation between absolute and relative uncertainty Δp = η(pp. With the use of the principle of uncertainty we can calculate Δx $\Delta x \,\cdot\, \Delta p \ge \frac{\hbar}{2}$ and we obtain

$\Delta x \ge \frac{\hbar}{2} \,\frac{1}{\Delta p} ,$ $\Delta x \ge \frac{\hbar}{2} \,\frac{1}{\eta(p)\cdot p} ,$ $\Delta x \ge \frac{\hbar}{2} \,\frac{1}{(\eta(m)+\eta(v))\cdot mv} ,$ $\Delta x \ge \frac{\hbar}{2} \,\frac{1}{\left(\frac{\Delta m}{m}+\frac{\Delta v}{v}\right)\cdot mv} ,$ $\Delta x \ge \frac{1.055\,\cdot\,10^{-34}}{2\,\cdot\,\left(\frac{1}{150}+\frac{1}{40}\right)\,\cdot\, 0.15\,\cdot\, 40}\ \mbox{m} ,$ $\Delta x \ge 2.8\,\cdot\,10^{-34}\ \mbox{m} .$

Not only does this uncertainty fall below the dimensions of a nucleus of the atom, it is also approaching the Planck-Wheeler length*. The space and time apparantely stop to exist in the dimensions smaller than this length, or rather they start to have completely different properties, which no one has been so far able to estimate, much less reasonably describe.

We can conclude that the restriction from the quantum mechanics principles of uncertainty doesn’t matter in this macroscopic situation.

* See http://en.wikipedia.org/wiki/Planck_length.