Plot of a <a href="https://en.wikipedia.org/wiki/Trajectory_of_a_projectile#Trajectory_of_a_projectile_with_air_resistance" class="extiw" title="en:Trajectory of a projectile">ballistic trajectory with air resistance</a> for a typical skydive from an airplane. The trajectory follows the differential equation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\ddot {\vec {r}}}(t)=-\mu \,|v|\cdot {\vec {v}}-g\cdot {\hat {\vec {y}}}}"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mover><mrow class="MJX-TeXAtom-ORD"><mover><mi>r</mi><mo stretchy="false">→<!-- → --></mo></mover></mrow><mo>¨<!-- ¨ --></mo></mover></mrow></mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−<!-- − --></mo><mi>μ<!-- μ --></mi><mspace width="thinmathspace"></mspace><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>v</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mo>⋅<!-- ⋅ --></mo><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mover><mi>v</mi><mo stretchy="false">→<!-- → --></mo></mover></mrow></mrow><mo>−<!-- − --></mo><mi>g</mi><mo>⋅<!-- ⋅ --></mo><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mover><mrow class="MJX-TeXAtom-ORD"><mover><mi>y</mi><mo stretchy="false">→<!-- → --></mo></mover></mrow><mo stretchy="false">^<!-- ^ --></mo></mover></mrow></mrow></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle {\ddot {\vec {r}}}(t)=-\mu \,|v|\cdot {\vec {v}}-g\cdot {\hat {\vec {y}}}}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb8722ff69e41eae085f768809133a42596347f5" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:22.806ex; height:3.509ex;" alt="{\displaystyle {\ddot {\vec {r}}}(t)=-\mu \,|v|\cdot {\vec {v}}-g\cdot {\hat {\vec {y}}}}"></span> with initial conditions <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {r}}(0)={\vec {0}},\;{\dot {\vec {r}}}(0)=v_{0}{\hat {\vec {x}}}}"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mover><mi>r</mi><mo stretchy="false">→<!-- → --></mo></mover></mrow></mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>=</mo><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mover><mn>0</mn><mo stretchy="false">→<!-- → --></mo></mover></mrow></mrow><mo>,</mo><mspace width="thickmathspace"></mspace><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mover><mrow class="MJX-TeXAtom-ORD"><mover><mi>r</mi><mo stretchy="false">→<!-- → --></mo></mover></mrow><mo>˙<!-- ˙ --></mo></mover></mrow></mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>=</mo><msub><mi>v</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mover><mrow class="MJX-TeXAtom-ORD"><mover><mi>x</mi><mo stretchy="false">→<!-- → --></mo></mover></mrow><mo stretchy="false">^<!-- ^ --></mo></mover></mrow></mrow></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle {\vec {r}}(0)={\vec {0}},\;{\dot {\vec {r}}}(0)=v_{0}{\hat {\vec {x}}}}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46813badc19f3a500e6d49f172344ad3c39df9df" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:20.974ex; height:3.509ex;" alt="{\displaystyle {\vec {r}}(0)={\vec {0}},\;{\dot {\vec {r}}}(0)=v_{0}{\hat {\vec {x}}}}"></span>.<br><p>The parameters are:
</p>
<ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g=9.81\,\mathrm {m/s^{2}} }"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mi>g</mi><mo>=</mo><mn>9.81</mn><mspace width="thinmathspace"></mspace><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">m</mi><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><msup><mi mathvariant="normal">s</mi><mrow class="MJX-TeXAtom-ORD"><mn>2</mn></mrow></msup></mrow></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle g=9.81\,\mathrm {m/s^{2}} }</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9158f614705e7145da785f4319777247422caa7e" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:13.805ex; height:3.176ex;" alt="{\displaystyle g=9.81\,\mathrm {m/s^{2}} }"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{\mathrm {max} }=50\,\mathrm {m/s} }"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><msub><mi>v</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow></mrow></msub><mo>=</mo><mn>50</mn><mspace width="thinmathspace"></mspace><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">m</mi><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mi mathvariant="normal">s</mi></mrow></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle v_{\mathrm {max} }=50\,\mathrm {m/s} }</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa9373f7a607b77c5c9eb959a27689ed7923b06b" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:14.244ex; height:2.843ex;" alt="{\displaystyle v_{\mathrm {max} }=50\,\mathrm {m/s} }"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu =k/m=g/v_{\mathrm {max} }^{2}}"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mi>μ<!-- μ --></mi><mo>=</mo><mi>k</mi><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mi>m</mi><mo>=</mo><mi>g</mi><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><msubsup><mi>v</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow></mrow><mrow class="MJX-TeXAtom-ORD"><mn>2</mn></mrow></msubsup></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle \mu =k/m=g/v_{\mathrm {max} }^{2}}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03e3e25200fd5030d2e150394b9a5254b3a4a43e" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:18.71ex; height:3.009ex;" alt="{\displaystyle \mu =k/m=g/v_{\mathrm {max} }^{2}}"></span></li>
<li>The horizontal starting velocity takes the values <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{0}=0\,\mathrm {m/s} }"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><msub><mi>v</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>=</mo><mn>0</mn><mspace width="thinmathspace"></mspace><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">m</mi><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mi mathvariant="normal">s</mi></mrow></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle v_{0}=0\,\mathrm {m/s} }</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13c14da05ba7b748cecec6dec1684f4976f8fc16" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:10.844ex; height:2.843ex;" alt="{\displaystyle v_{0}=0\,\mathrm {m/s} }"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{0}=17\,\mathrm {m/s} }"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><msub><mi>v</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>=</mo><mn>17</mn><mspace width="thinmathspace"></mspace><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">m</mi><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mi mathvariant="normal">s</mi></mrow></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle v_{0}=17\,\mathrm {m/s} }</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb760ba9267a337ba11600c119dc9da1f3503131" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:12.007ex; height:2.843ex;" alt="{\displaystyle v_{0}=17\,\mathrm {m/s} }"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{0}=33\,\mathrm {m/s} }"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><msub><mi>v</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>=</mo><mn>33</mn><mspace width="thinmathspace"></mspace><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">m</mi><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mi mathvariant="normal">s</mi></mrow></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle v_{0}=33\,\mathrm {m/s} }</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/515ebbf69a3001daa6a52b6f77aa737c7b4bf802" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:12.007ex; height:2.843ex;" alt="{\displaystyle v_{0}=33\,\mathrm {m/s} }"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{0}=50\,\mathrm {m/s} }"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><msub><mi>v</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>=</mo><mn>50</mn><mspace width="thinmathspace"></mspace><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">m</mi><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mi mathvariant="normal">s</mi></mrow></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle v_{0}=50\,\mathrm {m/s} }</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d949ddf98d802cfc32d453390835cf19db461bf" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:12.007ex; height:2.843ex;" alt="{\displaystyle v_{0}=50\,\mathrm {m/s} }"></span></li></ul>
The second-order ordinary differential equation system is <a href="https://en.wikipedia.org/wiki/Ordinary_differential_equation#Reduction_to_a_first-order_system" class="extiw" title="w:Ordinary differential equation">reduced to a first-order system</a>, using velocity components as independent variables. The system is then solved numerically using <a rel="nofollow" class="external text" href="http://docs.scipy.org/doc/scipy/reference/generated/scipy.integrate.odeint.html#scipy.integrate.odeint">Scipy odeint</a>.
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