## Related questions with answers

Why is the following situation impossible? An air rifle is used to shoot 1.00-g particles at a speed of $v_{x}=100 \mathrm{m} / \mathrm{s} .$ The rifle's barrel has a diameter of 2.00 mm. The rifle is mounted on a perfectly rigid support so that it is fired in exactly the same way each time. Because of the uncertainty principle, however, after many firings, the diameter of the spray of pellets on a paper target is 1.00 cm.

Solution

VerifiedFirst let's express the $\textbf{uncertainty principle}$ for the y direction:

$\begin{align*} \Delta p_{y} \Delta a&=\dfrac{h}{4\pi} \end{align*}$

Since we know the diameter of the cannon for sure we can write:

$\begin{align*} \Delta p_{y} a&=\dfrac{h}{4\pi}\\ \implies \Delta p_{y}&=\dfrac{h}{4 a \pi}\\ \end{align*}$

Now if we use some trigonometry:

$\begin{align*} tan\theta&=\dfrac{d}{l} \end{align*}$

Where l is the distance to the target. That can also be expressed as:

$\begin{align*} tan\theta&=\dfrac{p_{y}}{p_{x}} \end{align*}$

Therefore:

$\begin{align*} l&=\dfrac{p_{x}d}{p_{y}}\\ \implies l&=\dfrac{p_{x}d}{\dfrac{h}{4 a \pi}}\\ \implies l&=\dfrac{p_{x}d 4 a \pi }{h}\\ \implies l&=\dfrac{0.001\ \text{kg}\cdot 100\ \text{m}\text{s}^{-1}\cdot 0.01\ \text{m}\cdot 4 \cdot 0.002\ \text{m}\cdot \pi }{6.626\cdot 10^{-34}\ \text{kg}\text{m}^{2}\text{s}^{-1}}\\ \implies l&=\boxed{3.79\cdot 10^{28}\ \text{m}} \end{align*}$

The distance is simply to large for this to be a possible scenario.

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