Animated Solution for Chemistry - Atomic Structure: A ball weighing $10\text{ g}$ is moving with a velocity of $90{\text{ ms}}^{-1}$. If the uncertainity in its velocity is $5\%$, then the uncertainty in its position is ......... $\times 10^{-33}\text{ m}$ (Rounded off to the nearest integer). [Given, $h=6.63\times 10^{-34}\text{ J-s}$]

The Everyday World

Have you ever wondered why we don't see baseballs teleporting or cars existing in multiple places at once? The answer lies in the sheer scale of the quantum world compared to our everyday macroscopic reality. In this problem, we are going to apply a purely quantum mechanical principle—Heisenberg's Uncertainty Principle—to a macroscopic object: a $10\text{ g}$ ball.

By the end of this calculation, you will mathematically prove to yourself why quantum fuzziness is completely invisible in our daily lives.

We are given a ball with a mass of $10\text{ g}$. In physics and chemistry, our first instinct should always be to convert everything into standard SI units. Since Planck's constant ($h$) is given in Joules-seconds (and $1\text{ J}=1{\text{ kg m}}^2{\text{ s}}^{-2}$), we must convert the mass into kilograms.

The ball is moving with a velocity of $90{\text{ ms}}^{-1}$. However, we are told there is a $5\%$ uncertainty in this velocity. This means the velocity isn't a perfect $90{\text{ ms}}^{-1}$; it has a slight spread or "fuzziness". Let's calculate the absolute value of this uncertainty, denoted as $\Delta v$.

Heisenberg's Uncertainty Principle

Werner Heisenberg famously stated that it is impossible to simultaneously know both the exact position and the exact momentum of a particle. The mathematical formulation of this principle is:

Since momentum ($p$) is the product of mass and velocity ($p=mv$), the uncertainty in momentum for an object of constant mass is $\Delta p=m\Delta v$. Substituting this into our equation gives:

We want to find the uncertainty in position, $\Delta x$. To find the minimum possible uncertainty, we treat the inequality as an equation and rearrange it to solve for $\Delta x$:

Now, we carefully substitute all our known values into the rearranged equation.

Let's simplify the denominator first to avoid any silly calculation mistakes. Multiplying the numbers: $4\times 3.14=12.56$, and $12.56\times 4.5=56.52$. Don't forget the $10^{-2}$ from the mass!

Dividing $6.63$ by $0.5652$ gives us approximately $11.73$.

To match the format requested in the question ($\times 10^{-33}$), we shift the decimal point one place to the left:

The question asks us to round off to the nearest integer. The nearest integer to $1.173$ is simply $1$.

Our final answer is $1\times 10^{-33}\text{ m}$. Take a moment to appreciate how unimaginably small this number is. The nucleus of an atom is about $10^{-15}\text{ m}$ across. The uncertainty in the position of this $10\text{ g}$ ball is a billion billion times smaller than an atomic nucleus!

This beautifully demonstrates why Heisenberg's Uncertainty Principle is only relevant for subatomic particles like electrons, and why our macroscopic world appears perfectly deterministic.