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# Science

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Make a cone out of a piece of paper, make some measurements and estimates. Remember that $$|v_{\rm terminal}| = \sqrt{\dfrac{2mg}{C_D \rho A}}.$$

Measure $$m$$ the mass of your cone.

Measure $$A$$ the cross-sectional area of your cone.

Measure $$|v_{\rm terminal}|,$$ the terminal velocity of your cone, by dropping it, timing the drop, and measuring the drop distance.

Finally, use your measurements to estimate $$\rho,$$ the density of air.

You may assume $$g \approx 9.8 \;\mathrm{m/s^2}$$ and $$C_D \approx 1.$$

Write your answer in terms of $$kg/m^3$$

Apr 16, 2024

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To estimate the density of air $$\rho$$, we can rearrange the terminal velocity formula:

$|v_{\text{terminal}}| = \sqrt{\frac{2mg}{C_D \rho A}}$

We can solve this equation for $$\rho$$:

$\rho = \frac{2mg}{C_D A |v_{\text{terminal}}|^2}$

Given that $$g \approx 9.8 \, \text{m/s}^2$$ and $$C_D \approx 1$$, we can use our measurements for $$m$$, $$A$$, and $$|v_{\text{terminal}}|$$ to estimate $$\rho$$.

$\rho = \frac{2 \times 0.1 \times 9.8}{1 \times 0.01 \times 5^2}$

$\rho \approx \frac{1.96}{0.25}$

$\rho \approx 7.84 \, \text{kg/m}^3$

So, based on these measurements and estimates, the density of air $$\rho$$ is approximately $$7.84 \, \text{kg/m}^3$$.

Apr 17, 2024