Equations of Motion in Two and Three Dimensions

Formulas worksheet

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

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Equation of position in rectilinear uniform motion -x-axis

x = x_{0} + v \cdot t

Position equation in free fall

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

Equation of speed in free fall

v = - g \cdot t

Equation of acceleration on the Earth surface

a = - g

Equation of acceleration in uniform rectilinear motion

a = 0

Equation of velocity in uniform rectilinear motion

v = v_{0} = cte

Projectile Motion

Equation of position x-axis

x = v_{x} \cdot t = v_{0} \cdot \cos (α) \cdot t

Equation of position y-axis

y = H + v_{0 y} \cdot t - \frac{1}{2} \cdot g \cdot t^{2} = H + v_{0} \cdot \sin (α) \cdot t - \frac{1}{2} \cdot g \cdot t^{2}

Equation of velocity x-axis

v_{x} = v_{0 x} = v_{0} \cdot \cos (α)

Equation of velocity y-axis

v_{y} = v_{0 y} - g \cdot t = v_{0} \cdot \sin (α) - g \cdot t

Equation of acceleration in the x-axis

a_{x} = 0

Equation of acceleration in the y-axis

a_{y} = - g

Relationship between the velocity in the X-Y axes and the angle of the trajectory

\tan (α) = \frac{c a t e t o o p u e s t o}{c a t e t o c o n t i g u o} = \frac{v_{y}}{v_{x}} \Rightarrow α = \tan^{- 1} (\frac{v_{y}}{v_{x}})

Angular Quantities

Linear space - angular space relationship

s = φ \cdot R

Average angular velocity

ω_{a} = \frac{∆ φ}{∆ t} = \frac{φ_{f} - φ_{i}}{t_{f} - t_{i}}

Angular velocity

ω = \lim_{∆ t \to 0} \frac{∆ φ}{∆ t} = \frac{d φ}{d t}

Average angular acceleration

α_{a} = \frac{∆ ω}{∆ t} = \frac{ω_{f} - ω_{i}}{t_{f} - t_{i}}

Angular acceleration

α = \lim_{∆ t \to 0} \frac{∆ ω}{∆ t} = \frac{d ω}{d t}

Normal acceleration equation (u.c.m. and u.a.c.m.)

a_{n} = \frac{v^{2}}{R} = ω^{2} \cdot R

Tangential acceleration - angular acceleration relationship

a_{t} = α \cdot R

Linear velocity - angular velocity relationship

v = ω \cdot R

Uniform Circular Motion (U.C.M.)

Position vector in circular motion

\vec{r} = x \cdot \vec{i} + y \cdot \vec{j} = R \cdot \cos (φ) \cdot \vec{i} + R \cdot \sin (φ) \cdot \vec{j}

Equation of tangential acceleration (u.c.m.)

a_{t} = 0

Angular velocity in u.c.m.

ω = constant

Angular acceleration (u.c.m.)

α = 0

Equations of Uniform Circular Motion (U.C.M.)

Angular position in u.c.m.

φ = φ_{0} + ω \cdot t

Angular velocity in u.c.m.

ω = constant

Angular acceleration (u.c.m.)

α = 0

Frequency in uniform circular motion (u.c.m.)

f = \frac{ω}{2 \cdot π}

Relationship angular velocity - period - frequency in u.c.m.

ω = \frac{2 \cdot π}{T} = 2 \cdot π \cdot f

Uniform Circular Motion (U.C.M.) Graphs

Angular position in u.c.m.

φ = φ_{0} + ω \cdot t

Angular velocity in u.c.m.

ω = constant

Angular acceleration (u.c.m.)

α = 0

Definition of tangent of an angle

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

Non-Uniform Circular Motion

Position vector in circular motion

\vec{r} = x \cdot \vec{i} + y \cdot \vec{j} = R \cdot \cos (φ) \cdot \vec{i} + R \cdot \sin (φ) \cdot \vec{j}

Angular position in non-u.c.m.

φ = φ_{0} + ω \cdot t + \frac{1}{2} \cdot α \cdot t^{2}

Angular velocity in non-u.c.m.

ω = ω_{0} + α \cdot t

Angular acceleration in non-u.c.m.

α = constant

Equations of Non-Uniform Circular Motion

Angular position in non-u.c.m.

φ = φ_{0} + ω \cdot t + \frac{1}{2} \cdot α \cdot t^{2}

Angular velocity in non-u.c.m.

ω = ω_{0} + α \cdot t

Angular acceleration in non-u.c.m.

α = constant

Non-Uniform Circular Motion Graphs

Angular position in non-u.c.m.

φ = φ_{0} + ω \cdot t + \frac{1}{2} \cdot α \cdot t^{2}

Angular velocity in non-u.c.m.

ω = ω_{0} + α \cdot t

Angular acceleration in non-u.c.m.

α = constant

2-D kinematics

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Formulas worksheet

Equation of position in rectilinear uniform motion -x-axis

Position equation in free fall

Equation of speed in free fall

Equation of acceleration on the Earth surface

Equation of acceleration in uniform rectilinear motion

Equation of velocity in uniform rectilinear motion

Equation of position x-axis

Equation of position y-axis

Equation of velocity x-axis

Equation of velocity y-axis

Equation of acceleration in the x-axis

Equation of acceleration in the y-axis

Relationship between the velocity in the X-Y axes and the angle of the trajectory

Linear space - angular space relationship

Average angular velocity

Angular velocity

Average angular acceleration

Angular acceleration

Normal acceleration equation (u.c.m. and u.a.c.m.)

Tangential acceleration - angular acceleration relationship

Linear velocity - angular velocity relationship

Position vector in circular motion

Equation of tangential acceleration (u.c.m.)

Angular velocity in u.c.m.

Angular acceleration (u.c.m.)

Angular position in u.c.m.

Angular velocity in u.c.m.

Angular acceleration (u.c.m.)

Frequency in uniform circular motion (u.c.m.)

Relationship angular velocity - period - frequency in u.c.m.

Angular position in u.c.m.

Angular velocity in u.c.m.

Angular acceleration (u.c.m.)

Definition of tangent of an angle

Position vector in circular motion

Angular position in non-u.c.m.

Angular velocity in non-u.c.m.

Angular acceleration in non-u.c.m.

Angular position in non-u.c.m.

Angular velocity in non-u.c.m.

Angular acceleration in non-u.c.m.

Angular position in non-u.c.m.

Angular velocity in non-u.c.m.

Angular acceleration in non-u.c.m.