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Kinematics


Remember that: a=a \vec{a} = \mathbf{a} , a˙=dadt\dot{a}= \frac{da}{dt} and Δa=afinalainitial=afai=a2a1\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 aa
distance/range dd
frequency ff
gravitational acceleration g=9.81m/s2g = 9.81 m/s^{2}
height hh
displacement rr
distance ss
period TT
time tt
velocity vv
initial horizontal coordinate xx
initial vertical coordinate yy
angular acceleration α\alpha
ratio of a circle's circumference to its diameter π=3.14159\pi = 3.14159
launch angle, angular displacement θ\theta
angular displacement ϕ\phi
angular velocity ω\omega

Uniform linear motion

name equation
average velocity vˉ=ΔrΔt\mathbf{\bar{v}}=\frac{\Delta \vec{r}}{\Delta t}
instantaneous velocity v=limΔt0ΔrΔt=drdt=r˙\vec{v}=\underset{\Delta t \to 0}\lim \frac{\Delta \vec{r}}{\Delta t} = \frac{d\vec{r}}{dt} = \mathbf{\dot{r}}
speed v=v=dsdtv = \vert \vec{v} \vert = \frac{ds}{dt}

Uniformly accelerated linear motion

name equation
average acceleration aˉ=ΔvΔt=vfvit\mathbf{\bar{a}}=\frac{\Delta \vec{v}}{\Delta t} = \frac{v_{f} - v_{i}}{t}
instantaneous acceleration a=limΔt0ΔvΔt=dvdt=v˙=r¨\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 Δr=vit+at22\Delta \vec{r} = \vec{v}_{i}t + \frac{\vec{a}t^{2}}{2}
position from velocity and time Δr=(vf+vi)t2\Delta \vec{r} = \frac{\left( \vec{v}_{f} + \vec{v}_{i}\right) t}{2}
position from acceleration and velocity r=vf2vi22ar = \frac{v_{f}^{2} - v_{i}^{2}}{2a}

Free fall

In this case a=g=9.81 m/s2 a = g = 9.81\ \mathrm{m/s^{2}}

name equation
height for time tt h=12gt2h = \frac{1}{2}gt^{2}
time taken from height hh t=2hgt = \sqrt{\frac{2h}{g}}
instantaneous velocity for time tt v=gtv = gt
instantaneous velocity for height hh v=2ghv = \sqrt{2gh}
average velocity for time tt vˉ=12gt\bar{v} = \frac{1}{2} gt
average velocity for height hh v=2gh2v = \frac{\sqrt{2gh}}{2}

Upward motion

In this case a=g=9.81 m/s2 a = g = 9.81\ \mathrm{m/s^{2}}

name equation
final velocity for time tt vf=vigtv_{f} = v_{i} - gt
final velocity for height hh vf2=vi22ghv_{f}^{2} = v_{i}^{2} - 2gh
distance when final velocity is unknown h=vitgt22h = v_{i}t - \frac{gt^{2}}{2}
maximum height hmax=vi22gh_{max} = \frac{v_{i}^{2}}{2g}
time for maximum height thmax=vigt_{h_{max}} = \frac{v_{i}}{g}
time of flight tmax=2vigt_{max} = \frac{2 v_{i}}{g}

Projectile motion

In this case a=g=9.81 m/s2 a = g = 9.81\ \mathrm{m/s^{2}}

name equation
speed horizontal component at t=0t = 0 v0x=v0cos(θ) v_{0_{x}} = v_{0} \cos\left( \theta \right)
speed vertical component at t=0t = 0 v0y=v0sin(θ) v_{0_{y}} = v_{0} \sin\left( \theta \right)
acceleration horizontal component ax=0 a_{x} = 0
acceleration vertical component ay=g a_{y} = -g
speed vertical component at any time vx=v0cos(θ) v_{x} = v_{0} \cos\left( \theta \right)
speed vertical component at any time vx=v0sin(θ)gt v_{x} = v_{0} \sin\left( \theta \right) - gt
velocity magnitude v=vx2+vy2v = \sqrt{v_{x}^{2} + v_{y}^{2}}
horizontal displacement x=v0tcos(θ)x = v_{0} t \cos\left( \theta \right)
vertical displacement y=v0tsin(θ)12gt2y = v_{0} t \sin\left( \theta \right) - \frac{1}{2}gt^{2}
displacement magnitude Δr=x2+y2\Delta r = \sqrt{x^{2} + y^{2}}
range dd for initial height y0y_{0} d=vcosθg(vsinθ+v2sin2θ+2gy0)d = \frac{v \cos \theta}{g} \left( v \sin \theta + \sqrt{v^{2} \sin^{2} \theta + 2gy_{0}} \right)
range if y0=0 y_{0} = 0 d=v2sin2θgd = \frac{v^{2} \sin 2\theta}{g}
initial velocity v0=x2gxsin2θ2ycos2θ v_{0} = \sqrt{\frac{x^{2}g}{x \sin 2\theta - 2y \cos^{2}\theta}}
time of flight t=2v0tsin(θ)g t = \frac{2v_{0}t \sin \left( \theta \right)}{\vert g \vert}
time of flight for height y0y_{0} t=dvcosθ=vsinθ+(vsinθ)2+2gy0g 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=xv0cos(θ) t = \frac{x}{v_{0} \cos \left( \theta \right)}
time to reach maximum height thmax=v0sin(θ)g t_{h_{max}} = \frac{v_{0} \sin \left( \theta \right)}{\vert g \vert}
maximum height hmax=v02sin2(θ)2g h_{max} = \frac{v_{0}^{2} \sin^{2} \left( \theta \right)}{2 \vert g \vert}
maximum height for known (x,y)\left(x,y\right) position and angle (θ)\left( \theta \right) hmax=(xtanθ)24(xtanθy) h_{max} = \frac{\left( x \tan \theta \right)^{2}}{4 \left( x \tan \theta - y \right)}
relation between dd and hmaxh_{max} hmax=dtanθ4h_{max} = \frac{d \tan \theta}{4}
maximum distance (θ=45C) \left( \theta = 45^{\circ}\mathrm{C} \right) dmax=v2g d_{max} = \frac{v^{2}}{\vert g \vert}
angle of reach for shallow trajectory θ=12arcsin(gdv2)\theta = \frac{1}{2} \arcsin \left( \frac{gd}{v^{2}} \right)
angle of reach for steep trajectory θ=12arccos(gdv2)\theta = \frac{1}{2} \arccos \left( \frac{gd}{v^{2}} \right)
angle required to hit a coordinate (x,y) \left( x, y \right) θ=arctan(v2±v4g(gx2+2yv2)gx) \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=v02g(sinθ+cos2θtanh1(sinθ))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=2πωT = \frac{2\pi}{\omega}
frequency f=1T=ω2πf = \frac{1}{T} = \frac{\omega}{2\pi}
angular velocity ω=2πT=2πf=dϕdt \omega = \frac{2 \pi}{T} = 2 \pi f = \frac{d\phi}{dt}
tangential velocity vt=2πTr=ωrv_{t} = \frac{2 \pi}{T}r = \omega r
angle ϕ \phi swept out in a time tt ϕ=2πTt=ωt\phi = \frac{2\pi}{T} t = \omega t

Uniformly accelerated circular motion

name equation
angular acceleration α=ωfωit \alpha = \frac{\omega_{f} - \omega_{i}}{t}
angle when the final angular velocity is unknown ϕ=ωit+αt22\phi = \omega_{i}t + \frac{\alpha t^{2}}{2}
angle from angular velocity and time ϕ=(ωf+ωi)t2\phi = \frac{\left(\omega_{f} + \omega_{i}\right) t}{2}
angle from angular acceleration and angular velocity ϕ=ωf2ωi22α\phi = \frac{\omega_{f}^{2} - \omega_{i}^{2}}{2\alpha}
tangential acceleration at=αra_{t} = \alpha r