# Unit 1: Kinematics

Formula Note
$$v = \frac{d}{\Delta t}$$ Calculate constant velocity.
$$v_f = v_i +at$$ Change in velocity is equal to acceleration x time.
$$\Delta d = v_it + \frac{1}{2}at^2$$ Quadratic equation for object undergoing acceleration.
$$t=\frac{-V_i \pm \sqrt{V_i^2+2a \Delta d}}{a}$$ Quadratic formula, time when projectile hits the ground.
$$\Delta d = \frac{v_i+v_f}{2}$$ Find distance travelled based on acerage velocity.
$$t=\sqrt{\frac{2 \Delta d}{a}}$$ If initial velocity is zero, easy way to solve for time.
$$v_f^2 = v_i^2+2a \Delta d$$ Square proportionality between both velocities, acceleration and distance travelled.

# Unit 2: Dynamics

Formula Note
$$a = \frac{F_{net}}{m}$$ Newton's second law.
$$F_g = mg$$ This is weight. 'g' is the gravitational constant, which is why you weigh differently on the moon!
$$F_N = -mg$$ Normal force, acts perpendicular to weight. Example, exerted by table on book to keep it on the table.
$$F_f = \mu F_N$$ Friction force, acts opposite to motion of an object.
$$F_s = -k \Delta x$$ Spring force which is proportional to negative displacement of an object since it tries to reach equilibrium.
$$\mu_s = \tan \theta$$ Find static coefficient of friction of an object on an inclined plane, theta being the angle when the object begins to slide.
$$F_x = mg \sin \theta$$ Force parralel to incline plane, moving the object.
$$F_y = mg \cos \theta$$ Force perpendicular to incline plane.
$$F_{on1by2} = -F_{on2by1}$$ Newton's third law of motion.
$$W = F \Delta d$$ Work done by an object.
$$W = F \Delta d \cos \theta$$ Work done by an object force is acting at an angle.
$$F_T = ma = Ma$$ Frictionless pulley with two objects attached on a string.
$$a = \frac{M_g - m_g}{m+M}$$ Find acceleration of objects on a pulley.
$$PE_g = mgh$$ Potential gravitational energy.
$$KE = \frac{1}{2}mv^2$$ Kinetic energy.
$$PE_{el} = \frac{1}{2}k \Delta x^2$$ Potential elastic energy.
$$W = \Delta KE + \Delta PE_g + \Delta PE_e$$ Change in mechanical energy is work.
$$h = L(1-\cos \theta)$$ Vertical height of pendulum string.
$$P = \frac{W}{t} = Fv$$ Power of an object.
$$e = \frac{P_{out}}{P_{in}}$$ Efficiency of a mechanical system.
$$p = mv$$ Momentum of an object.
$$F = \frac{\Delta p}{\Delta t} = m(\frac{v_f-v_i}{t})$$ Newton's second law.
$$\Delta p = F \Delta t$$ Impulse-momentum formula.
$$m_1v_{i1} + m_2v_{i2} = m_1v_{f1} + m_2v_{f2}$$ Perfectly elastic collisions.
$$m_1v_1 + m_2v_2 = v_f(m_1+m_2)$$ Perfectly inelastic collisions. Objects move at same speed before or after collision.

# Unit 3: Thermodynamics

Formula Note
$$Q = mc \Delta T$$ Energy, or heat, in a medium of set mass.
$$Q_{latent} = mL$$ Latent heat of an object as it goes through phase changes.
$$Q_1 = -Q_2$$ Heat gained by one object is lost by the other object.
$$PV=nRT$$ Relationship between pressure, volume, and temperature in an ideal gas.
$$PV = NK_BT$$ Same as previous formula but using Boltzman's constant.
$$\frac{P_1V_1}{T_1}=\frac{P_2V_2}{T_2}$$ Derivation of what happens to a a set object when it changes pressure, temperature, or volume.
$$P=\frac{F}{A}$$ Formula for pressure acting on a given surface.
$$PV = \frac{N}{3}mv^2$$ Theoretical equation of an ideal gas.
$$KE = \frac{3}{2}K_BT$$ Average kinetic energy of molecules in an ideal gas.
$$v = \sqrt{\frac{3K_BT}{m}}$$ Mean value of speed of molecules in an ideal gas.

# Unit 4: Waves

Formula Note
$$\Delta x = x_0 \cos (\omega t)$$ Displacement of an oscillating object.
$$\omega = 2 \pi f$$ Speed of wave.
$$\omega^2 = \frac{k}{m}$$ Speed of wave in an elastic string.
$$T = \frac{1}{f} \rightarrow f = \frac{1}{T}$$ Relationship between frequency and time.
$$a = -{\omega}^2x$$ Current acceleration.
$$a_{max} = -{\omega}x_0$$ Maximum acceleration.
$$v_{max} = \omega x_0$$ Maximum velocity.
$$v(t) = -x_0 \omega \sin (\omega t)$$ Velocity as a function of time.
$$a(t) = x_0 \cos (\omega t)$$ Acceleration as a function of time.
$$KE(t) = \frac{1}{2}mx_0^2 {\omega}^2 \sin (\omega t)$$ Kinetic energy as a function of time.
$$PE_{elastic}(t) = \frac{1}{2}kx_0^2 {\cos}^2 (\omega t)$$ Potential elastic energy as a function of time.
$$\lvert \Delta d \rvert = n \lambda$$ Path difference for constructive interference of double slit.
$$\lvert \Delta d \rvert = (n+\frac{1}{2}) \lambda$$ Path difference for destructive interference of double slit.
$$d \sin \theta = n \lambda$$ Local maximum of intensity acting on a wall after a double slit.
$$\frac{S}{D} = \frac{n \lambda}{d}$$ Small angle approximation derived from previous formula.
$$\theta_i = \theta_r$$ Angle of incident is equal to angle of reflection.
$$n_1 \sin \theta_i = n_2 \sin \theta_r$$ Snell's law for angle of refraction.
$$\sin \theta_c = \frac{n_2}{n_1}$$ Snell's law when $\theta_r = 90°$ for critical angle of refraction.
$$n = \frac{c}{v}$$ Refractive index of a wave. Higher index means wave travels slower in a given medium.
$$b \sin \theta = \frac{n \lambda}{b}$$ Destructive interference in a single slit.
$$I \propto A^2$$ Intensity is related to the square of the amplitude of the wave.
$$I \propto x^{-2}$$ If distance from source increases, intensity decreases by the square of that.
$$I = I_0 \cos^2 \theta$$ Intensity of a polarized light source, maximum at $n\pi, n \in \mathbb{N}$.

# Unit 5: Electricity and Magnetism

Formula Note
$$F = k \frac{q_1q_2}{r^2}$$ Coulomb's Law. The electric force between two charges, seperated by distance r. What's $k$, you may ask?
$$k = \frac{1}{4\pi\varepsilon_0} \approx 8.99 Nm^2C^{-2}$$ Coulomb constant, defined in terms of the permittivity of free space.
$$W=Vq$$ Work done (energy) by a particle is equal to the voltage times the charge.
$$E = \frac{F}{q_2} = \frac{q_1}{r^2}$$ Electric field strength acting on a charge. Divide by the charge of which you are finding the field strength.
$$I = nAvq \rightarrow v = \frac{I}{nAq}$$ More details on current in terms of the velocity of the electrons, the right formula will come in handy to calculate their speed!
$$I = \frac{\Delta q}{\Delta t}$$ Current is the rate of change of charge (how fast/many electrons are flowing).
$$V = IR$$ Ohm's Law. Beautiful. In an ohmic system, voltage is the product of current times resistance.
$$\Sigma V = 0$$ Kirchhoff's voltage law, sum of all voltages across any loop in a circuit will be 0.
$$\Sigma I = 0$$ Kirchhoff's current law, sum of all current intering a node equals the sum of all current leaving the node.
$$P = VI = I^2R = \frac{V^2}{R}$$ Power dissipated in a source of resistance (joule heating)
$$R_{total} = \Sigma R = R_1 + R_2 + ...$$ Sum resistance when resistors are IN SERIES (same wire).
$$R_{total} = \frac{1}{\Sigma \frac{1}{R}} = \frac{1}{\frac{1}{R_1} + \frac{1}{R_2} + ...}$$ Sum reciprocal resistance when resistors are IN PARALLEL (seperated by node).
$$\delta = \frac{RA}{L} = \frac{R\pi r^2}{L}$$ Resistivity equals resistance times cross sectional area of wire divided by length of wire.
$$\varepsilon = I(R + r) \rightarrow V = \varepsilon - Ir$$ Internal resistance of a battery cell, notably the output voltage DECREASES when current INCREASES.
$$F = qvB \sin \theta$$ Lorentz force for a particle in a magnetic field of charge $q$ and velocity $v$, where $\theta$ is the angle to the normal (they will try and trick you with this, force is highest when at a right angle!)
$$F = BIL \sin \theta$$ Lorentz force for wire of length $L$ and current $I$, where $\theta$ is the angle to the normal (they will try and trick you with this, force is highest when at a right angle!)