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>. 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)=-k\,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>k</mi><mspace width="thinmathspace"></mspace><mi>v</mi><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)=-k\,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/ee57015f1dddc11e40e392d5983faef1cc5e7690" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:21.322ex; height:3.509ex;" alt="{\displaystyle {\ddot {\vec {r}}}(t)=-k\,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}}}\cos(\alpha _{0})+{\hat {\vec {y}}}\sin(\alpha _{0}))}"><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><mo stretchy="false">(</mo><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><mi>cos</mi><mo>⁡<!-- ⁡ --></mo><mo stretchy="false">(</mo><msub><mi>α<!-- α --></mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo stretchy="false">)</mo><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><mi>sin</mi><mo>⁡<!-- ⁡ --></mo><mo stretchy="false">(</mo><msub><mi>α<!-- α --></mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle {\vec {r}}(0)={\vec {0}},\;{\dot {\vec {r}}}(0)=v_{0}({\hat {\vec {x}}}\cos(\alpha _{0})+{\hat {\vec {y}}}\sin(\alpha _{0}))}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87df81771068ab54e4de96a2cb1572367c8e9633" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:42.36ex; height:3.509ex;" alt="{\displaystyle {\vec {r}}(0)={\vec {0}},\;{\dot {\vec {r}}}(0)=v_{0}({\hat {\vec {x}}}\cos(\alpha _{0})+{\hat {\vec {y}}}\sin(\alpha _{0}))}"></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=1}"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mi>g</mi><mo>=</mo><mn>1</mn></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle g=1}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba0027652d8e5f694d4aa1c71ce16c9380ce1186" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:5.377ex; height:2.509ex;" alt="{\displaystyle g=1}"></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 k=1}"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mi>k</mi><mo>=</mo><mn>1</mn></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle k=1}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c035ffa69b5bca8bf2d16c3da3aaad79a8bcbfa" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:5.472ex; height:2.176ex;" alt="{\displaystyle k=1}"></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 \alpha _{0}=45^{\circ }}"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><msub><mi>α<!-- α --></mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>=</mo><msup><mn>45</mn><mrow class="MJX-TeXAtom-ORD"><mo>∘<!-- ∘ --></mo></mrow></msup></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle \alpha _{0}=45^{\circ }}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2c2b72e37866ae57af89bf8e8b5a2f4a5efdd1b" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:9.019ex; height:2.676ex;" alt="{\displaystyle \alpha _{0}=45^{\circ }}"></span></li>
<li>The initial 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}=2}"><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>2</mn></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle v_{0}=2}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80c6c36a9aaded04fafb488db42bc92594e827d2" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:6.443ex; height:2.509ex;" alt="{\displaystyle v_{0}=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_{0}=4}"><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>4</mn></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle v_{0}=4}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0813bb1e87e76c30cbfbdcc192eeccedbaf2d4a2" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:6.443ex; height:2.509ex;" alt="{\displaystyle v_{0}=4}"></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}=6}"><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>6</mn></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle v_{0}=6}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18e5ce654cc4115d49c3837ab6c9ec96fa04a073" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:6.443ex; height:2.509ex;" alt="{\displaystyle v_{0}=6}"></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}=8}"><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>8</mn></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle v_{0}=8}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bcaae2efea8e2a2342464549a69efb37dbbf520" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:6.443ex; height:2.509ex;" alt="{\displaystyle v_{0}=8}"></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}=10}"><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>10</mn></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle v_{0}=10}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f1e0692ab93b6f5f8f18495c8dec5179747d83e" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:7.605ex; height:2.509ex;" alt="{\displaystyle v_{0}=10}"></span></li></ul>
The differential equation is 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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