Uncertainty Principle and dart of mass

[teme92]

Homework Statement

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.

Homework Equations

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?

 

[soarce]

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

 

[teme92]

Hey soarce I edited the omission.

 

[soarce]

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