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# Assignments

## Drag (Wed 16 Oct 2024)

Consider the drag force on a falling cone of mass $m$, radius $r$, and half-angle
$\theta$.  In class we considered the model where the drag force depends on the speed
of the falling cone $v$, with the drag force being related to the change in the vertical
momentum of the air molecules, hence depending on the mass density of the air $\rho$.

For a single air molecule, the change in vertical momentum, assuming an elastic
collision, is
\begin{gather*}
  \Delta p_{\text{molecule}} = 2m_{\text{air}}v \sin\theta.
\end{gather*}
This gave an estimate of the total drag in terms of the number of molecules $N$ deflected
per unit time $\Delta t$: 
\begin{gather*}
  F_{\text{drag}} \approx \frac{N}{\Delta t} \Delta p_{\text{molecule}},\qquad
  N \approx A n v \Delta t,\qquad
  A = \pi r^2.
\end{gather*}
This gives
\begin{gather*}
  F_{\text{drag}} \approx 2A v^2 \underbrace{m_{\text{air}} n}_{\rho_{\text{air}}} \sin\theta.
\end{gather*}
Terminal velocity will be reached when this balances gravity $F_{\text{drag}} = mg$
giving a terminal velocity of
\begin{gather*}
  v_{\text{terminal}} \approx \sqrt{\frac{mg}{2A\rho_{\text{air}}\sin\theta}}.
\end{gather*}

An alternative hypothesis is that the drag force depends only on the cross-sectional
area $A$ and that there should therefore be no factor of $\sin\theta$ in the terminal
velocity.


### The Assignment

Review this derivation, correcting it as needed, then estimate whether or not we will be
able to test the hypothesis of the $\sin\theta$ dependence in class.  What other factors
have we neglected?  What are the relevant magnitudes?  How might the appear in the data?
How much data will we need to take?

Design an experiment to be completed in Wednesday's class.