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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Horizontal Launch

Equation of position in rectilinear uniform motion -x-axis

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

Position equation in free fall

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

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

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

Projectile Motion

Equation of position x-axis

$x={v}_{x}\cdot t={v}_{0}·\mathrm{cos}\left(\alpha \right)·t$

Equation of position y-axis

$y=\mathrm{H}+{\mathrm{v}}_{0\mathrm{y}}·\mathrm{t}-\frac{1}{2}·g·{t}^{2}=\mathrm{H}+{\mathrm{v}}_{0}·\mathrm{sin}\left(\mathrm{\alpha }\right)·\mathrm{t}-\frac{1}{2}·g·{t}^{2}$

Equation of velocity x-axis

${v}_{x}={v}_{0x}={v}_{0}·\mathrm{cos}\left(\alpha \right)$

Equation of velocity y-axis

${v}_{y}={v}_{0y}-g\cdot t={v}_{0}·\mathrm{sin}\left(\alpha \right)-g\cdot t$

Equation of acceleration in the x-axis

${a}_{x}=0$

Equation of acceleration in the y-axis

${a}_{y}=-g$

Angular Quantities

Linear space - angular space relationship

$s=\phi ·R$

Average angular velocity

${\omega }_{a}=\frac{∆\phi }{∆t}=\frac{{\phi }_{f}-{\phi }_{i}}{{t}_{f}-{t}_{i}}$

Angular velocity

$\omega =\underset{∆t\to 0}{\mathrm{lim}}\frac{∆\phi }{∆t}=\frac{d\phi }{dt}$

Average angular acceleration

${\alpha }_{a}=\frac{∆\omega }{∆t}=\frac{{\omega }_{f}-{\omega }_{i}}{{t}_{f}-{t}_{i}}$

Angular acceleration

$\alpha =\underset{∆t\to 0}{\mathrm{lim}}\frac{∆\omega }{∆t}=\frac{d\omega }{dt}$

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

${a}_{n}=\frac{{v}^{2}}{R}={\omega }^{2}·R$

Tangential acceleration - angular acceleration relationship

${a}_{t}=\alpha ·R$

Linear velocity - angular velocity relationship

$v=\omega ·R$

Uniform Circular Motion (U.C.M.)

Position vector in circular motion

$\stackrel{\to }{r}=x·\stackrel{\to }{i}+y·\stackrel{\to }{j}=R·\mathrm{cos}\left(\phi \right)·\stackrel{\to }{i}+R·\mathrm{sin}\left(\phi \right)·\stackrel{\to }{j}$

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

${a}_{t}=0$

Angular velocity in u.c.m.

$\omega =\text{constant}$

Angular acceleration (u.c.m.)

$\alpha =0$

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

Angular position in u.c.m.

$\phi ={\phi }_{0}+\omega ·t$

Angular velocity in u.c.m.

$\omega =\text{constant}$

Angular acceleration (u.c.m.)

$\alpha =0$

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

$f=\frac{\omega }{2·\pi }$

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

$\omega =\frac{2·\pi }{T}=2·\pi ·f$

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

Angular position in u.c.m.

$\phi ={\phi }_{0}+\omega ·t$

Angular velocity in u.c.m.

$\omega =\text{constant}$

Angular acceleration (u.c.m.)

$\alpha =0$

Non-Uniform Circular Motion

Position vector in circular motion

$\stackrel{\to }{r}=x·\stackrel{\to }{i}+y·\stackrel{\to }{j}=R·\mathrm{cos}\left(\phi \right)·\stackrel{\to }{i}+R·\mathrm{sin}\left(\phi \right)·\stackrel{\to }{j}$

Angular position in non-u.c.m.

$\phi ={\phi }_{0}+\omega ·t+\frac{1}{2}·\alpha ·{t}^{2}$

Angular velocity in non-u.c.m.

$\omega ={\omega }_{0}+\alpha ·t$

Angular acceleration in non-u.c.m.

$\alpha =\text{constant}$

Equations of Non-Uniform Circular Motion

Angular position in non-u.c.m.

$\phi ={\phi }_{0}+\omega ·t+\frac{1}{2}·\alpha ·{t}^{2}$

Angular velocity in non-u.c.m.

$\omega ={\omega }_{0}+\alpha ·t$

Angular acceleration in non-u.c.m.

$\alpha =\text{constant}$

Non-Uniform Circular Motion Graphs

Angular position in non-u.c.m.

$\phi ={\phi }_{0}+\omega ·t+\frac{1}{2}·\alpha ·{t}^{2}$

Angular velocity in non-u.c.m.

$\omega ={\omega }_{0}+\alpha ·t$

Angular acceleration in non-u.c.m.

$\alpha =\text{constant}$