# Uncertainty Principle and dart of mass

1. Aug 6, 2015

### teme92

1. The problem statement, all variables and given/known data
A dart of mass $m$ is dropped from a height $l$. Formulate the uncertainty principle and estimate the minimum limitations, set by the uncertainty principle, of the accuracy that can be achieved in the lateral $x$ position after falling, given an original uncertainty $\Delta x$, and no original uncertainty in y.

2. Relevant equations

3. The attempt at a solution
The time to drop is $\sqrt{2l/g}$. With uncertainty the dart hits the ground at a distance $R$ where:

$R=\Delta x+\sqrt{2l/g}\space \Delta p/m \geq\Delta x +\sqrt{2l/g}\space \hbar/2m\Delta x$

This next part I found from a book but I don't understand how its got from above:

$\Delta x=(l\hbar^2/2m^2g)^{\frac{1}{4}}$

with $R\geq(8l\hbar^2/m^2g)^{\frac{1}{4}}$

Could anyone clarify how $\Delta x$ and $R$ were found?

Last edited: Aug 6, 2015
2. Aug 6, 2015

### soarce

Please review the statement of the problem, it seems that the last sentence is not finished and maybe we miss important data.

3. Aug 6, 2015

### teme92

Hey soarce I edited the omission.

4. Aug 6, 2015

### soarce

Your derivation seems correct. Can you give us the reference from where you took the formula for $\Delta x$?