Name

Equations

Units (if specified)

Extra Notes

Average speed

$\overline{v}=\dfrac{d}{\Delta t}$

$m/s$

Average velocity

$\overline{v}=\dfrac{\Delta x}{\Delta t}$

$m/s$

2D kinematics: replace x with r

Instantaneous velocity

$\overrightarrow{v}=\lim _{\Delta t\rightarrow 0}\dfrac{\Delta \overrightarrow{x}}{\Delta t}$

$m/s$

2D kinematics: replace x with r

Average acceleration

$\overrightarrow{\overline{a}}=\dfrac{\Delta \overrightarrow{v}}{\Delta t}$

$m/s^{2}$

Displacement

$\Delta x=x-x_{0}$

$m$

2D kinematics: replace x with r

Slope

$\dfrac{\Delta y}{\Delta x}$

$\dfrac{\Delta x}{\Delta t}$ or $\dfrac{\Delta v}{\Delta x}$

Kinematics (1D)

$v=v_{0}+at$

$x=\dfrac{1}{2}\left( v_{0}+v\right) t$

$v^{2}=v_{0}^{2}+2ax$

$x=v_{0}t+\dfrac{1}{2}at^{2}$

Kinematics (2D)

$v_{?}=v_{0?}+a_{?}t$

$?=v_{0?}t+\dfrac{1}{2}a_{?}t^{2}$

$?=\dfrac{1}{2}\left( v_{0?}+v_{?}\right) t$

$v_{?}^{2}=v_{0?}^{2}+2a_{?}?$

? means replace with x OR y

Projectile motion:

$a_{y}=-9.80m/s^{2}$

$a_{x}=0$

Relative velocity

$v_{ab}=v_{b}-v_{a}$

$m/s$

Newton’s 2nd Law

$\Sigma F=ma$

N

2D: $\sum F_{?}=ma_{?}$

Gravitational force

$F=G\dfrac{m_{1}m_{2}}{r^{2}}$

$\dfrac{N\cdot m^{2}}{kg^{2}}$

$G =6.673\times 10^{-11}$

$N\cdot m^{2}/kg^{2}$

Weight

$W=mg$

N

Always acts down

Normal force

$F_{N}=mg$

N

Perpendicular force

Static friction

$f_{s}^{MAX}=\mu_s \cdot F_{N}$

Force before breakaway

Kinetic friction

$f_{\text{k}} = \mu_k \cdot F_{N}$

Moving surfaces

Velocity in circular motion

$v_{c}=\dfrac{2\pi r}{T}$

$m/s$

Changes direction, not constant

Period (T)

$T=\dfrac{1}{frequency}$

$s$

To make one revolution

Centripetal acceleration

$a_{c}=\dfrac{v^{2}}{r}$

$m/s^{2}$

Direction towards the center

Centripetal force

$F_{c}=\dfrac{mv^{2}}{r}$ or $F_{c}=ma_{c}$

N

Always directed towards the center, changes direction

Speed of a satellite

$v=\sqrt{\dfrac{GM_{E}}{r}}$

$m/s$

Period of a satellite

$T=\dfrac{2\pi r^{\dfrac{3}{2}}}{\sqrt{GM_{E}}}$

Work

$W=Fs$ or

$W=Fcos\theta s$

J

cos0° = 1 (F)

cos90° = 0

cos180° = -1 (-F)

Kinetic energy

$KE=\dfrac{1}{2}mv^{2}$

J

Always positive

Work-energy theorem

$W=KE_{f}-KE_{0}$

J

Potential energy and gravitational PE

$PE=mgh$

$W_{gravity}=mg\left(h_{0}-h_{f}\right)$

J

Stored energy

PE max = KE is 0

PE is 0 = KE max

Work for closed path

$W_{gravity}=0 J$

J

Conservative and Nonconservative forces acting together

$W=W_{c}+W_{nc}$

J

Both forces act on an object at the same time

Conservation of energy

$E_{f}=E_{0}$

$KE_{f}+PE_{f}=KE_{0}+PE_{0}$

J

Work energy theorem (nonconservative)

$W_{nc}=E_{f}-E_{0}$ or

$\left( PE_{f}+KE_{f}\right) -\left( PE_{0}+KE_{0}\right)$

J

Total mechanical energy

$E=KE+PE$

J

Average power

$\overline{P}=\dfrac{W}{t}$

W

Impulse

$\overrightarrow{J}=F\Delta t$

$N\cdot s$

change in momentum

same direction as avg. force

Linear momentum

$\overrightarrow{p}=m\overrightarrow{v}$

$kg\cdot m/s$

mass in motion

Impulse-momentum theorem

$\sum F\Delta t=p_{f}-p_{0}$

when a net force acts on an object, the impulse force is equal to the change in the momentum

Principle of conservation of linear momentum

$p_{f}=p_{0}$

Collisions in 1D

$p_{f1}+p_{f2}=p_{01}+p_{02}$

Collisions in 2D

$p_{f1?}+p_{f2?}=p_{01?}+p_{02?}$

? means x or y

Center of mass (CoM)

$x_{cm}=\dfrac{m_{1}x_{1}+m_{2}x_{2}}{m_{1}+m_{2}}$

Velocity of CoM

$v_{cm}=\dfrac{m_{1}v_{1}+m_{2}v_{2}}{m_{1}+m_{2}}$

Angular displacement

$\Delta \theta =\theta -\theta _{0}$

Angular displacement in radians

$\theta =\dfrac{s}{r}$

Arc length

$s=r\theta$

Full revolution

$2\pi rad=360°$

Angular velocity

$\overline{\omega }=\dfrac{\Delta \theta }{\Delta t}$

Angular acceleration

$\overline{\alpha }=\dfrac{\Delta \omega }{\Delta t}$

Rotational kinematics

$\omega =\omega _{0}+\alpha t$

$\theta =\dfrac{1}{2}\left( \omega _{0}+\omega \right) t$

$\theta =\omega _{0}t+\dfrac{1}{2}\alpha t^{2}$

$\omega ^{2}=\omega _{0}^{2}+2\alpha \theta$

Tangential velocity

$v_{T}=r\omega$

Tangential acceleration

$a_{T}=r\alpha$

Centripetal acceleration

$a_{c}=r\omega ^{2}$

Pythagorean theorem

(acceleration)

$\overrightarrow{a}=\sqrt{\overrightarrow{a}_{c}^{2}+\overrightarrow{a_{T}}^{2}}$

Translational velocity

$v=rw$

Translational acceleration

$a=r\alpha$

Torque

$\tau =Fl$

Equilibrium conditions

$\Sigma F_{x}=0$

$\Sigma F_{y}=0$

$\Sigma \tau =0$

Center of gravity

$x_{cg}=\dfrac{W_{1}x_{1}+W_{2}x_{2}}{W_{1}+W_{2}}$

Tangential force

$F_{T}=ma_{T}$

w/ sub $\tau =\left( mr^{2}\right) \alpha$

Inertia

$I=mr^{2}$

Moment of inertia

$\Sigma \tau=\sum \left( mr^{2}\right) \alpha$

$\tau1+\tau2+\tau3$

Net external force

$\Sigma \tau =I\alpha$

Rotational work

$W_{R}=\tau \theta$

Rotational KE

$KE_{R}=\dfrac{1}{2}I\omega ^{2}$

Total KE

$KE_{tota1}=\dfrac{1}{2}I\omega ^{2}+\dfrac{1}{2}mv_{T}^{2}$

Mechanical energy of a rotating object

$E=\dfrac{1}{2}mv^{2}+\dfrac{1}{2}I\omega ^{2}+mgh$

Energy conservation: $E_{f}=E_{0}$

Angular momentum

$L=I\omega$

Mass density

$\rho =\dfrac{m}{V}$

Pressure

$P=\dfrac{F}{A}$

Pressure and depth in static fluid

$P_{2}=P_{1}+\rho gh$

Pressure gauges

$P_{atm}=\rho gh$

Archimede’s principle

$F_{B}=\rho Vg$

buoyant force = mass of displaced fluid

Linear thermal expansion

$\Delta L=\alpha L_{0}\Delta T$

Volume thermal expansion

$\Delta V=\beta V_{0}\Delta T$

Specific heat capacity

$Q=mc\Delta T$

Latent heat

$Q=mL$

Restoring force produced by a spring

$F_{x}=kx$

Hooke’s law

$F_{x}=-kx$

displacement

$x=A\cos \omega t$

Frequency

$f=\dfrac{1}{T}$

Angular frequency

$\omega =2\pi f$ or

$\omega =\sqrt{\dfrac{k}{m}}$

Max speed (shm)

$v_{max}=Aw$

Max acceleration (shm)

$a_{max}=Aw^{2}$

Work done by spring

$W=\left( F\cos \theta \right) s$

$W=\dfrac{1}{2}kx_{0}^{2}-\dfrac{1}{2}kx_{f}^{2}$

Mechanical energy

insert

PE elastic

$\dfrac{1}{2}kx^{2}$