## 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.

Click on the icon    to export them to any external compatible program.

### 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

$a=-g$