12.9: Summary
- Page ID
- 92193
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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 \(\theta\).