Question

A ball weighing $10\mathrm{g}$ is moving with a velocity of $90{\mathrm{ms}}^{-1}$. If the uncertainty in its velocity is $5\%$, then the uncertainty in its position is ________×${10}^{-33}\mathrm{m}$.

$[\mathrm{Given}:\mathrm{h}=6.63\times {10}^{-34}\mathrm{J}\mathrm{s}]$

Answer 1

Step 1: Given data

Mass of ball, $\mathrm{m}=10\mathrm{g}=10\times {10}^{-3}\mathrm{Kg}$

Velocity of ball, $\mathrm{v}=90{\mathrm{ms}}^{-1}$

Uncertainty in velocity$=5\%$

Therefore, $∆\mathrm{v}=\mathrm{v}\times \frac{5}{100}∆\mathrm{v}=90\times \frac{5}{100}=4.5{\mathrm{ms}}^{-1}$

Step 2: Calculating the uncertainty in the position

According to the Heisenberg's uncertainty principle, we have:

$∆\mathrm{x}\times ∆\mathrm{p}=\frac{\mathrm{h}}{4\mathrm{\pi }}$

$\text{'}∆\mathrm{x}\text{'}$ is the uncertainty in the position

$\text{'}∆\mathrm{p}\text{'}$ is the uncertainty in the momentum

$\text{'}\mathrm{h}\text{'}$is the Planck's constant

Now using the above principle we get:

$$\begin{gathered} ∆\mathrm{x}\times ∆\mathrm{p}=\frac{\mathrm{h}}{4\mathrm{\pi }} \\ \Rightarrow ∆\mathrm{x}\times \mathrm{m}∆\mathrm{v}=\frac{\mathrm{h}}{4\mathrm{\pi }} \\ \Rightarrow ∆\mathrm{x}\times \frac{10}{1000}\times 4.5=\frac{6.626\times {10}^{-34}}{4\mathrm{\pi }} \\ \Rightarrow ∆\mathrm{x}=\frac{6.626\times {10}^{-34}\times 7\times 1000}{4\times 22\times 4.5} \\ \Rightarrow ∆\mathrm{x}=1.17\times {10}^{-33} \\ \Rightarrow ∆\mathrm{x}=\mathrm{x}\times {10}^{-33}\mathrm{m} \end{gathered}$$

Hence, $\mathrm{x}=1.17\simeq 1$.Therefore uncertainty in the position is equal to $1\times {10}^{-33}\mathrm{m}$.