Formulas worksheet

Here is a full formulary by subject Motion in Physics. By understanding each equation, you will be able to solve any problem that you may find at this level.

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Position Vector

Modulus position vector in Cartesian coordinates in 3 dimensions

$\left|\stackrel{\to }{r}\right|=\sqrt{{x}^{2}+{y}^{2}+{z}^{2}}$

Modulus position vector in Cartesian coordinates in 2 dimensions

$\left|\stackrel{\to }{r}\right|=\sqrt{{x}^{2}+{y}^{2}}$

Position vector in 2 dimensional Cartesian coordinates

$\stackrel{\to }{r}=x\stackrel{\to }{i}+y\stackrel{\to }{j}$

Position vector in 3 dimensional Cartesian coordinates

$\stackrel{\to }{r}=x\stackrel{\to }{i}+y\stackrel{\to }{j}+z\stackrel{\to }{k}$

Trajectory and Equation of Position

Position equation in two dimensional Cartesian

$\stackrel{\to }{r}\left(t\right)=x\left(t\right)\stackrel{\to }{i}+y\left(t\right)\stackrel{\to }{j}$

Position equation in three dimensional Cartesian

$\stackrel{\to }{r}\left(t\right)=x\left(t\right)\stackrel{\to }{i}+y\left(t\right)\stackrel{\to }{j}+z\left(t\right)\stackrel{\to }{k}$

Displacement

Displacement vector in three dimensions, Cartesian coordinates

$∆\stackrel{\to }{r}={\stackrel{\to }{r}}_{f}-{\stackrel{\to }{r}}_{i}=\left({x}_{f}-{x}_{i}\right)\stackrel{\to }{i}+\left({y}_{f}-{y}_{i}\right)\stackrel{\to }{j}+\left({z}_{f}-{z}_{i}\right)\stackrel{\to }{k}$

Magnitude of the displacement vector in two-dimensional Cartesian

$\left|∆\stackrel{\to }{r}\right|=\sqrt{{\left({x}_{f}-{x}_{i}\right)}^{2}+{\left({y}_{f}-{y}_{i}\right)}^{2}}$

Displacement vector in two dimensions, Cartesian coordinates

$∆\stackrel{\to }{r}={\stackrel{\to }{r}}_{f}-{\stackrel{\to }{r}}_{i}=\left({x}_{f}-{x}_{i}\right)\stackrel{\to }{i}+\left({y}_{f}-{y}_{i}\right)\stackrel{\to }{j}$

Magnitude of the displacement vector in three-dimensional Cartesian

$\left|∆\stackrel{\to }{r}\right|=\sqrt{{\left({x}_{f}-{x}_{i}\right)}^{2}+{\left({y}_{f}-{y}_{i}\right)}^{2}+{\left({z}_{f}-{z}_{i}\right)}^{2}}$

Distance Traveled

Circumference arc length

$L=\theta \cdot r$

Instantaneous Velocity

Magnitude velocity 3 Cartesian dimensions

$\left|\stackrel{\to }{v}\right|=\sqrt{{v}_{x}^{2}+{v}_{y}^{2}+{v}_{z}^{2}}$

Velocity magnitude 2 Cartesian dimensions

$\left|\stackrel{\to }{v}\right|=\sqrt{{v}_{x}^{2}+{v}_{y}^{2}}$

Instantaneous Speed

Instantaneous speed

$V =\underset{∆t\to 0}{\mathrm{lim}}\frac{∆s}{∆t}$

Average Acceleration

Average Acceleration

${\stackrel{\to }{a}}_{a}=\frac{{\stackrel{\to }{v}}_{2}-{\stackrel{\to }{v}}_{1}}{{t}_{2}-{t}_{1}}=\frac{∆\stackrel{\to }{v}}{∆t}$

Instantaneous Acceleration

Acceleration magnitude in 3 Cartesian dimensions

$\left|\stackrel{\to }{a}\right|=\sqrt{{a}_{x}^{2}+{a}_{y}^{2}+{a}_{z}^{2}}$

Acceleration magnitude in 2 Cartesian dimensions

$\left|\stackrel{\to }{a}\right|=\sqrt{{a}_{x}^{2}+{a}_{y}^{2}}$

Intrinsic Components of Acceleration

Magnitude of the acceleration as a function of intrinsic components

$\left|\stackrel{\to }{a}\right|=\sqrt{{a}_{t}^{2}+{a}_{n}^{2}}$

Acceleration as a function of the intrinsic components

$\stackrel{\to }{a}={\stackrel{\to }{a}}_{t}+{\stackrel{\to }{a}}_{n}={a}_{t}{\stackrel{\to }{u}}_{t}+{a}_{n}{\stackrel{\to }{u}}_{n}$

Tangential Acceleration

Tangential acceleration

${\stackrel{\to }{a}}_{t}=\frac{dv}{dt}{\stackrel{\to }{u}}_{t}$

Normal or Centripetal Acceleration

Normal or centripetal acceleration

${\stackrel{\to }{a}}_{n}=\frac{{v}^{2}}{\rho }{\stackrel{\to }{u}}_{n}$

Equations of Constant Velocity Motion

Equation of position in rectilinear uniform motion -x-axis

$x={x}_{0}+v\cdot t$

Equation of acceleration in uniform rectilinear motion

$a=0$

Equation of velocity in uniform rectilinear motion

$v={v}_{0}=\text{cte}$

Constant Velocity Motion Graphs

Equation of position in rectilinear uniform motion -x-axis

$x={x}_{0}+v\cdot t$

Equation of acceleration in uniform rectilinear motion

$a=0$

Equation of velocity in uniform rectilinear motion

$v={v}_{0}=\text{cte}$

Equations of Constant Acceleration Motion

Position equation of uniformly accelerated rectilinear motion - x-axis

$x={x}_{0}+{v}_{0}t+\frac{1}{2}a{t}^{2}$

Position equation of uniformly accelerated rectilinear motion - y-axis

$y={y}_{0}+{v}_{0}t+\frac{1}{2}a{t}^{2}$

Acceleration equation in uniformly accelerated rectilinear motion

$a=\text{cte}$

Velocity equation in uniformly accelerated rectilinear motion

$v={v}_{0}+a\cdot t$

Constant Acceleration Motion Graphs

Position equation of uniformly accelerated rectilinear motion - x-axis

$x={x}_{0}+{v}_{0}t+\frac{1}{2}a{t}^{2}$

Velocity equation in uniformly accelerated rectilinear motion

$v={v}_{0}+a\cdot t$

Acceleration equation in uniformly accelerated rectilinear motion

$a=\text{cte}$

Free Fall

Position equation in free fall

$y=\mathrm{H}-\frac{1}{2}g{t}^{2}$

Position equation of uniformly accelerated rectilinear motion - y-axis

$y={y}_{0}+{v}_{0}t+\frac{1}{2}a{t}^{2}$

Equation of speed in free fall

$v=-g\cdot t$

Equation of acceleration on the Earth surface

$a=-g$

Vertical Launch

Equation of position of downward vertical launch

$y=\mathrm{H}-{v}_{0}t-\frac{1}{2}g{t}^{2}$

Equation of position in upward vertical launch

$y=\mathrm{H}+{v}_{0}t-\frac{1}{2}g{t}^{2}$

Position equation of uniformly accelerated rectilinear motion - y-axis

$y={y}_{0}+{v}_{0}t+\frac{1}{2}a{t}^{2}$

Equation of velocity of downward vertical launch vertical

$v=-{v}_{0}-g\cdot t$

Equation of velocity of the upward vertical launch

$v={v}_{0}-g\cdot t$

Equation of acceleration on the Earth surface

$a=-g$