Kinematics
Remember that: \( \vec{a} = \mathbf{a} \), \(\dot{a}= \frac{da}{dt}\) and \(\Delta a = a_{final} - a_{initial} = a_{f} - a_{i} = a_{2} - a_{1} \). The \(\propto\) symbol is read as "is proportional to".
Symbols
| name | symbol |
|---|---|
| acceleration | $$a$$ |
| distance | $$d$$ |
| frequency | $$f$$ |
| gravitational acceleration | $$g = 9.81\ \mathrm{m/s}^{2}$$ |
| height | $$h$$ |
| range | $$R$$ |
| displacement | $$r$$ |
| distance | $$s$$ |
| period/time of flight | $$T$$ |
| time | $$t$$ |
| velocity | $$v$$ |
| initial horizontal coordinate | $$x$$ |
| initial vertical coordinate | $$y$$ |
| angular acceleration | $$\alpha$$ |
| ratio of a circle's circumference to its diameter | $$\pi = 3.14159$$ |
| launch angle, angular displacement | $$\theta$$ |
| angular displacement | $$\phi$$ |
| angular velocity | $$\omega$$ |
Uniform linear motion
| name | equation |
|---|---|
| average velocity | $$\mathbf{\bar{v}}=\frac{\Delta \vec{r}}{\Delta t}$$ |
| instantaneous velocity | $$\vec{v}=\underset{\Delta t \to 0}\lim \frac{\Delta \vec{r}}{\Delta t} = \frac{d\vec{r}}{dt} = \mathbf{\dot{r}}$$ |
| speed | $$v = \vert \vec{v} \vert = \frac{ds}{dt}$$ |
Uniformly accelerated linear motion
| name | equation |
|---|---|
| average acceleration | $$\mathbf{\bar{a}}=\frac{\Delta \vec{v}}{\Delta t} = \frac{v_{f} - v_{i}}{t}$$ |
| instantaneous acceleration | $$\vec{a}=\underset{\Delta t \to 0}\lim \frac{\Delta \vec{v}}{\Delta t} = \frac{d\vec{v}}{dt} = \mathbf{\dot{v}} = \mathbf{\ddot{r}}$$ |
| position when final velocity is unknown | $$\Delta \vec{r} = \vec{v}_{i}t + \frac{\vec{a}t^{2}}{2}$$ |
| position from velocity and time | $$\Delta \vec{r} = \frac{\left( \vec{v}_{f} + \vec{v}_{i}\right) t}{2}$$ |
| position from acceleration and velocity | $$r = \frac{v_{f}^{2} - v_{i}^{2}}{2a}$$ |
Free fall
In this case \( a = g = 9.81\ \mathrm{m/s^{2}} \)
| name | equation |
|---|---|
| height for time \(t\) | $$h = \frac{1}{2}gt^{2}$$ |
| time taken from height \(h\) | $$t = \sqrt{\frac{2h}{g}}$$ |
| instantaneous velocity for time \(t\) | $$v = gt$$ |
| instantaneous velocity for height \(h\) | $$v = \sqrt{2gh}$$ |
| average velocity for time \(t\) | $$\bar{v} = \frac{1}{2} gt$$ |
| average velocity for height \(h\) | $$v = \frac{\sqrt{2gh}}{2}$$ |
Upward motion
In this case \( a = g = 9.81\ \mathrm{m/s^{2}} \)
| name | equation |
|---|---|
| final velocity for time \(t\) | $$v_{f} = v_{i} - gt$$ |
| final velocity for height \(h\) | $$v_{f}^{2} = v_{i}^{2} - 2gh$$ |
| distance when final velocity is unknown | $$h = v_{i}t - \frac{gt^{2}}{2}$$ |
| maximum height | $$h_{\max} = \frac{v_{i}^{2}}{2g}$$ |
| time for maximum height | $$t_{h_{\max}} = \frac{v_{i}}{g}$$ |
| time of flight | $$T = \frac{2 v_{i}}{g}$$ |
Projectile motion
In this case \( a = g = 9.81\ \mathrm{m/s^{2}} \). If you want to check the derivations of the projectile motion equations for maximum height, range, and flight time, you can refer to the following link.
| name | equation |
|---|---|
| horizontal component of speed at \(t = 0\) | $$ v_{0_{x}} = v_{0} \cos\left( \theta \right) $$ |
| vertical component of speed at \(t = 0\) | $$ v_{0_{y}} = v_{0} \sin\left( \theta \right) $$ |
| horizontal component of acceleration | $$ a_{x} = 0 $$ |
| vertical component of acceleration | $$ a_{y} = -g $$ |
| horizontal component of speed at any time | $$ v_{x} = v_{0} \cos\left( \theta \right) $$ |
| vertical component of speed at any time | $$ v_{y} = v_{0} \sin\left( \theta \right) - gt $$ |
| velocity magnitude | $$v = \sqrt{v_{x}^{2} + v_{y}^{2}}$$ |
| horizontal displacement | $$x = v_{0} t \cos\left( \theta \right)$$ |
| vertical displacement | $$y = v_{0} t \sin\left( \theta \right) - \frac{1}{2}gt^{2}$$ |
| displacement magnitude | $$\Delta r = \sqrt{x^{2} + y^{2}}$$ |
| range if \( y_{0} = 0 \) | $$d = \frac{v^{2} \sin 2\theta}{g} $$ |
| range \(d\) for initial height \(y_{0}\) | $$d = \frac{v \cos \theta}{g} \left( v \sin \theta + \sqrt{v^{2} \sin^{2} \theta + 2gy_{0}} \right)$$ |
| initial velocity | $$ v_{0} = \sqrt{\frac{x^{2}g}{x \sin 2\theta - 2y \cos^{2}\theta}} $$ |
| time of flight | $$ T = \frac{2v_{0} \sin \left( \theta \right)}{\vert g \vert}$$ |
| time of flight for height \(y_{0}\) | $$ T = \frac{d}{v \cos \theta} = \frac{v \sin \theta + \sqrt{\left( v \sin \theta \right)^{2} + 2gy_{0}}}{g} $$ |
| time of flight to the target position | $$ T = \frac{x}{v_{0} \cos \left( \theta \right)} $$ |
| time to reach maximum height | $$ t_{h_{\max}} = \frac{v_{0} \sin \left( \theta \right)}{\vert g \vert} $$ |
| maximum height | $$ h_{\max} = \frac{v_{0}^{2} \sin^{2} \left( \theta \right)}{2 \vert g \vert} $$ |
| maximum height for known \(\left(x,y\right)\) position and angle \(\left( \theta \right)\) | $$ h_{\max} = \frac{\left( x \tan \theta \right)^{2}}{4 \left( x \tan \theta - y \right)} $$ |
| relation between \(d\) and \(h_{\max}\) | $$h_{\max} = \frac{d \tan \theta}{4}$$ |
| maximum distance \( \left( \theta = 45^{\circ}\mathrm{C} \right) \) | $$ d_{\max} = \frac{v^{2}}{\vert g \vert} $$ |
| angle of reach for shallow trajectory | $$\theta = \frac{1}{2} \arcsin \left( \frac{gd}{v^{2}} \right)$$ |
| angle of reach for steep trajectory | $$\theta = \frac{1}{2} \arccos \left( \frac{gd}{v^{2}} \right)$$ |
| angle required to hit a coordinate \( \left( x, y \right) \) | $$ \theta = \arctan \left( \frac{v^{2} \pm \sqrt{v^{4} - g\left( gx^{2} + 2yv^{2} \right)}}{gx} \right) $$ |
| total path length of the trajectory | $$L = \frac{v_{0}^{2}}{g} \left( \sin \theta + \cos^{2} \theta \cdot \tanh^{-1} \left( \sin \theta \right) \right)$$ |
Uniform circular motion
| name | equation |
|---|---|
| period | $$T = \frac{2\pi}{\omega}$$ |
| frequency | $$f = \frac{1}{T} = \frac{\omega}{2\pi}$$ |
| angular velocity | $$ \omega = \frac{2 \pi}{T} = 2 \pi f = \frac{d\phi}{dt} $$ |
| tangential velocity | $$v_{t} = \frac{2 \pi}{T}r = \omega r$$ |
| angle \( \phi \) swept out in a time \(t\) | $$\phi = \frac{2\pi}{T} t = \omega t$$ |
Uniformly accelerated circular motion
| name | equation |
|---|---|
| angular acceleration | $$ \alpha = \frac{\omega_{f} - \omega_{i}}{t} $$ |
| angle when the final angular velocity is unknown | $$\phi = \omega_{i}t + \frac{\alpha t^{2}}{2}$$ |
| angle from angular velocity and time | $$\phi = \frac{\left(\omega_{f} + \omega_{i}\right) t}{2}$$ |
| angle from angular acceleration and angular velocity | $$\phi = \frac{\omega_{f}^{2} - \omega_{i}^{2}}{2\alpha} $$ |
| tangential acceleration | $$a_{t} = \alpha r$$ |