Equations of Motion in Physics

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

|\vec{r}| = \sqrt{x^{2} + y^{2} + z^{2}}

Modulus position vector in Cartesian coordinates in 2 dimensions

|\vec{r}| = \sqrt{x^{2} + y^{2}}

Position vector in 2 dimensional Cartesian coordinates

\vec{r} = x \vec{i} + y \vec{j}

Position vector in 3 dimensional Cartesian coordinates

\vec{r} = x \vec{i} + y \vec{j} + z \vec{k}

Trajectory and Equation of Position

Position equation in two dimensional Cartesian

\vec{r} (t) = x (t) \vec{i} + y (t) \vec{j}

Position equation in three dimensional Cartesian

\vec{r} (t) = x (t) \vec{i} + y (t) \vec{j} + z (t) \vec{k}

Displacement

Displacement vector in three dimensions, Cartesian coordinates

∆ \vec{r} = {\vec{r}}_{f} - {\vec{r}}_{i} = (x_{f} - x_{i}) \vec{i} + (y_{f} - y_{i}) \vec{j} + (z_{f} - z_{i}) \vec{k}

Magnitude of the displacement vector in two-dimensional Cartesian

|∆ \vec{r}| = \sqrt{{(x_{f} - x_{i})}^{2} + {(y_{f} - y_{i})}^{2}}

Displacement vector in two dimensions, Cartesian coordinates

∆ \vec{r} = {\vec{r}}_{f} - {\vec{r}}_{i} = (x_{f} - x_{i}) \vec{i} + (y_{f} - y_{i}) \vec{j}

Magnitude of the displacement vector in three-dimensional Cartesian

|∆ \vec{r}| = \sqrt{{(x_{f} - x_{i})}^{2} + {(y_{f} - y_{i})}^{2} + {(z_{f} - z_{i})}^{2}}

Distance Traveled

Circumference arc length

L = θ \cdot r

Average Velocity

Average velocity

{\vec{v}}_{a v g} = \frac{∆ \vec{r}}{∆ t} = \frac{\vec{r_{2}} - \vec{r_{1}}}{t_{2} - t_{1}}

Instantaneous Velocity

Magnitude velocity general expression

|\vec{v}| = \lim_{∆ t \to 0} |{\vec{v}}_{a v g}| = \lim_{∆ t \to 0} \frac{|∆ \vec{r}|}{∆ t} = \lim_{∆ t \to 0} \frac{∆ s}{∆ t}

Magnitude velocity 3 Cartesian dimensions

|\vec{v}| = \sqrt{v_{x}^{2} + v_{y}^{2} + v_{z}^{2}}

Velocity magnitude 2 Cartesian dimensions

|\vec{v}| = \sqrt{v_{x}^{2} + v_{y}^{2}}

Instantaneous velocity

\vec{v} = \lim_{∆ t \to 0} {\vec{v}}_{a v g} = \lim_{∆ t \to 0} \frac{∆ \vec{r}}{∆ t} = \frac{d \vec{r}}{d t}

Average Speed

Average speed

V_{avg} = \frac{∆ s}{∆ t} = \frac{s_{2} - s_{1}}{t_{2} - t_{1}}

Instantaneous Speed

Instantaneous speed

V = \lim_{∆ t \to 0} \frac{∆ s}{∆ t}

Average Acceleration

{\vec{a}}_{a} = \frac{{\vec{v}}_{2} - {\vec{v}}_{1}}{t_{2} - t_{1}} = \frac{∆ \vec{v}}{∆ t}

Instantaneous Acceleration

Acceleration magnitude in 3 Cartesian dimensions

|\vec{a}| = \sqrt{a_{x}^{2} + a_{y}^{2} + a_{z}^{2}}

Acceleration magnitude in 2 Cartesian dimensions

|\vec{a}| = \sqrt{a_{x}^{2} + a_{y}^{2}}

Instantaneous acceleration. General form

\vec{a} = \lim_{∆ t \to 0} {\vec{a}}_{a} = \lim_{∆ t \to 0} \frac{∆ \vec{v}}{∆ t} = \frac{d \vec{v}}{d t}

Intrinsic Components of Acceleration

Magnitude of the acceleration as a function of intrinsic components

|\vec{a}| = \sqrt{a_{t}^{2} + a_{n}^{2}}

Acceleration as a function of the intrinsic components

\vec{a} = {\vec{a}}_{t} + {\vec{a}}_{n} = a_{t} {\vec{u}}_{t} + a_{n} {\vec{u}}_{n}

Tangential Acceleration

Tangential acceleration

{\vec{a}}_{t} = \frac{d v}{d t} {\vec{u}}_{t}

Normal or Centripetal Acceleration

Normal or centripetal acceleration

{\vec{a}}_{n} = \frac{v^{2}}{ρ} {\vec{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} = 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} = cte

Definition of tangent of an angle

\tan (α) = \frac{opposite cathetus}{adjacent cathetus} = \frac{b}{c}

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 = 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 = cte

Free Fall

Position equation in free fall

y = 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 = H - v_{0} t - \frac{1}{2} g t^{2}

Equation of position in upward vertical launch

y = 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

1-D kinematics

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Formulas worksheet

Modulus position vector in Cartesian coordinates in 3 dimensions

Modulus position vector in Cartesian coordinates in 2 dimensions

Position vector in 2 dimensional Cartesian coordinates

Position vector in 3 dimensional Cartesian coordinates

Position equation in two dimensional Cartesian

Position equation in three dimensional Cartesian

Displacement vector in three dimensions, Cartesian coordinates

Magnitude of the displacement vector in two-dimensional Cartesian

Displacement vector in two dimensions, Cartesian coordinates

Magnitude of the displacement vector in three-dimensional Cartesian

Circumference arc length

Average velocity

Magnitude velocity general expression

Magnitude velocity 3 Cartesian dimensions

Velocity magnitude 2 Cartesian dimensions

Instantaneous velocity

Average speed

Instantaneous speed

Average Acceleration

Acceleration magnitude in 3 Cartesian dimensions

Acceleration magnitude in 2 Cartesian dimensions

Instantaneous acceleration. General form

Magnitude of the acceleration as a function of intrinsic components

Acceleration as a function of the intrinsic components

Tangential acceleration

Normal or centripetal acceleration

Equation of position in rectilinear uniform motion -x-axis

Equation of acceleration in uniform rectilinear motion

Equation of velocity in uniform rectilinear motion

Equation of position in rectilinear uniform motion -x-axis

Equation of acceleration in uniform rectilinear motion

Equation of velocity in uniform rectilinear motion

Definition of tangent of an angle

Position equation of uniformly accelerated rectilinear motion - x-axis

Position equation of uniformly accelerated rectilinear motion - y-axis

Acceleration equation in uniformly accelerated rectilinear motion

Velocity equation in uniformly accelerated rectilinear motion

Position equation of uniformly accelerated rectilinear motion - x-axis

Velocity equation in uniformly accelerated rectilinear motion

Acceleration equation in uniformly accelerated rectilinear motion

Position equation in free fall

Position equation of uniformly accelerated rectilinear motion - y-axis

Equation of speed in free fall

Equation of acceleration on the Earth surface

Equation of position of downward vertical launch

Equation of position in upward vertical launch

Position equation of uniformly accelerated rectilinear motion - y-axis

Equation of velocity of downward vertical launch vertical

Equation of velocity of the upward vertical launch

Equation of acceleration on the Earth surface