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I am exploring how the cross sectional area of a drogue chute affects its velocity after travelling a set distance.

I got this equation:

$$m_1 a = m_2 g - \frac12 c_d \rho A v^2,$$

This covers the force that is pushing the object forward; a load attached to the object was dropped which is why the masses are different.

Since I can't use the equations for uniform acceleration, I am finding it hard to find an equation that relates velocity to $A$. I tried to substitute $a= \mathrm dv/\mathrm dt$ and $v =\mathrm dx/\mathrm dt$ but I didn't make any progress. Could anyone help me out.

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  • $\begingroup$ Do you know how to solve ordinary differential equations? $\endgroup$
    – Michael Seifert
    Commented Jul 8 at 11:31
  • $\begingroup$ Hi, I have made some edits to your question to try to improve the readability, in particular by adding some latex math formatting. A sentence was not clear to me, though, so please check that the question still reads as intended and check that all signs and symbols are as intended. Otherwise my edit can be reverted. $\endgroup$
    – Steeven
    Commented Jul 8 at 11:38

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The equation $$ \frac {d v}{dt}= A+ B\, v^2 $$ with constant $A$ and $B$ is an example of a Riccati equation. The Wikipedi article shows you how to solve it.

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  • $\begingroup$ Why do you replace $dv/dt$ by $v dv/dx$? That is sometimes useful, but not here. You can first find $v(t)$ and then integrate to get $x(t)$. $\endgroup$
    – mike stone
    Commented Jul 8 at 13:23
  • $\begingroup$ Hmm it makes sense now. Thank you so much everyone! $\endgroup$
    – Melon
    Commented Jul 8 at 13:30

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