12.9: Summary
- Page ID
- 92193
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)This following table summarizes the formulæ for a projectile initially at the origin, fired with initial velocity \(v_{0}\) at an angle \(\theta\) from the horizontal.
Table 9-1. Summary of formulæ for projectile motion.
| Quantity | Formula |
|---|---|
| \(x(t)\) | \(x=\left(v_{0} \cos \theta\right) t\) |
| \(y(t)\) | \(y=-\dfrac{1}{2} g t^2+\left(v_0 \sin \theta\right) t\) |
| \(y(x)\) | \(y(x)=\left(-\dfrac{g}{2 v_{0}^{2} \cos ^{2} \theta}\right) x^{2}+(\tan \theta) x\) |
| Time in flight | \(t_{f}=\dfrac{2}{g} v_{0} \sin \theta\) |
| Range at angle \(\theta\) | \(R=\dfrac{v_{0}^{2}}{g} \sin 2 \theta\) |
| Max. range (at \(\left.\theta=45^{\circ}\right)\) | \(R_{\max }=\dfrac{v_{0}^{2}}{g}\) |
| Angle needed to hit target at range \(R\) for fixed \(v_{0}\) | \(\theta=\dfrac{1}{2} \sin ^{-1}\left(\dfrac{g R}{v_{0}^{2}}\right)\) |
| Speed needed to hit target at range \(R\) for fixed \(\theta\) | \(v_{0}=\sqrt{\dfrac{g R}{\sin 2 \theta}}\) |
| Max. altitude | \(h=\dfrac{v_{0}^{2} \sin ^{2} \theta}{2 g}\) |
| Speed needed to hit target at \(\left(x_{t}, y_{t}\right)\) for fixed \(\theta\) | \(\left(x_{t}, y_{t}\right)\) for fixed \(\theta\) & \(v_{0}=\sqrt{\dfrac{g x_{t}}{2\left(\tan \theta-\dfrac{y_{t}}{x_{t}}\right) \cos ^{2} \theta}}\) |
| Angle needed to hit target at \(\left(x_{t}, y_{t}\right)\) for fixed \(v_{0}\) | \(x_{t} \sin 2 \theta-2 y_{t} \cos ^{2} \theta=\dfrac{g x_{t}^{2}}{v_{0}^{2}}\) |
Note that this last expression must be solved iteratively for \(\theta\).