Physics Formulas (Situational)

Given: Initial velocity $v_0$ , final velocity $v_f$ , and time $t$ . Find: Kinematic average displacement $\Delta x$ .

$$\Delta x = \frac{v_0 + v_f}{2}t$$

Given: Change in velocity $\Delta v$ and time $t$ . Find: Acceleration $a$ .

$$a = \frac{\Delta v}{t}$$

Given: Masses of multiple objects $m_1, m_2, \dots$ and their positions $x_1, x_2, \dots$ . Find: Center of mass $x_{cm}$ .

$$x_{cm} = \frac{m_1x_1 + m_2x_2 + \dots}{m_1 + m_2 + \dots}$$

Given: Force $F$ , lever arm distance $r$ , and angle $\theta$ . Find: Torque $\tau$ .

$$\tau = rF\sin(\theta)$$

Given: Output force $F_{out}$ and input force $F_{in}$ . Find: Mechanical advantage $MA$ .

$$MA = \frac{F_{out}}{F_{in}}$$

Given: Initial kinetic and potential energy $KE_i, PE_i$ . Find: Final kinetic and potential energy $KE_f, PE_f$ in a closed system.

$$KE_i + PE_i = KE_f + PE_f$$

Given: Kinetic energy $KE$ and potential energy $PE$ . Find: Total mechanical energy $E$ .

$$E = KE + PE$$

Given: Change in total mechanical energy $\Delta E$ . Find: Work done by non-conservative forces $W_{nc}$ .

$$W_{nc} = \Delta E$$

Given: Force $F_1$ applied to an area $A_1$ in a confined fluid. Find: Resulting force $F_2$ on a second area $A_2$ .

$$\frac{F_1}{A_1} = \frac{F_2}{A_2}$$

Given: Density of fluid $\rho$ , volume $V$ , and gravity $g$ . Find: Weight of the fluid $W_{fluid}$ .

$$W_{fluid} = \rho V g$$

Given: Force $F$ applied perpendicular to a surface area $A$ . Find: Pressure $P$ .

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

Given: Original length $L$ , coefficient of linear expansion $\alpha$ , and change in temperature $\Delta T$ . Find: Change in length $\Delta L$ .

$$\Delta L = \alpha L \Delta T$$

Given: Original volume $V$ , coefficient of volumetric expansion $\beta$ , and change in temperature $\Delta T$ . Find: Change in volume $\Delta V$ .

$$\Delta V = \beta V \Delta T$$

Given: Constant pressure $P$ and change in volume $\Delta V$ . Find: Work done by the gas $W$ .

$$W = P \Delta V$$

Given: Reversible heat transfer $Q$ at a constant temperature $T$ . Find: Change in entropy $\Delta S$ .

$$\Delta S = \frac{Q}{T}$$

Given: Amount of charge $\Delta q$ passing through a point over time $\Delta t$ . Find: Current $I$ .

$$I = \frac{\Delta q}{\Delta t}$$

Given: Voltage drops across individual components $V_1, V_2, \dots$ in a series circuit. Find: Total voltage $V_s$ .

$$V_s = V_1 + V_2 + \dots$$

Given: Voltage across one branch $V_1$ in a parallel circuit. Find: Total voltage $V_p$ .

$$V_p = V_1 = V_2 = \dots$$

Given: Capacitances $C_1, C_2, \dots$ connected in series. Find: Equivalent capacitance $C_{eq}$ .

$$\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + \dots$$

Given: Capacitances $C_1, C_2, \dots$ connected in parallel. Find: Equivalent capacitance $C_{eq}$ .

$$C_{eq} = C_1 + C_2 + \dots$$

Given: Voltage across plates $V$ and distance between them $d$ . Find: Electric field strength in a capacitor $E$ .

$$E = \frac{V}{d}$$

Given: Current $I$ in a long straight wire, permeability of free space $\mu_0$ , and distance $r$ from the wire. Find: Magnetic field $B$ .

$$B = \frac{\mu_0 I}{2\pi r}$$

Given: Current $I$ in a circular loop of radius $r$ , and the permeability of free space $\mu_0$ . Find: Magnetic field at the center of the loop $B$ .

$$B = \frac{\mu_0 I}{2r}$$

Given: Charge $q$ moving with velocity $v$ at an angle $\theta$ in a magnetic field $B$ . Find: Magnetic force $F_B$ .

$$F_B = qvB\sin(\theta)$$

Given: Current $I$ , length of wire $L$ , angle $\theta$ , and magnetic field $B$ . Find: Magnetic force on the wire $F_B$ .

$$F_B = ILB\sin(\theta)$$

Given: Frequency $f$ or period $T$ . Find: Angular frequency $\omega$ .

$$\omega = 2\pi f = \frac{2\pi}{T}$$

Given: Bulk modulus of the medium $B$ and its density $\rho$ . Find: Speed of sound $v$ .

$$v = \sqrt{\frac{B}{\rho}}$$

Given: Power of the source $P$ and the area it is spread over $A$ . Find: Wave intensity $I$ .

$$I = \frac{P}{A}$$

Given: Final intensity $I_f$ and initial intensity $I_i$ . Find: Change in sound level/decibels $\Delta \beta$ .

$$\Delta \beta = 10 \log\left(\frac{I_f}{I_i}\right)$$

Given: Length of the string/pipe $L$ and harmonic number $n = 1, 2, 3, \dots$ . Find: Wavelength for string and open pipes $\lambda$ .

$$\lambda = \frac{2L}{n}$$

Given: Length of the closed pipe $L$ and odd harmonic number $n = 1, 3, 5, \dots$ . Find: Wavelength for closed pipes $\lambda$ .

$$\lambda = \frac{4L}{n}$$

Given: Wave speed $v$ , length $L$ , and harmonic number $n = 1, 2, 3, \dots$ . Find: Frequency for string and open pipes $f$ .

$$f = \frac{nv}{2L}$$

Given: Wave speed $v$ , length $L$ , and odd harmonic number $n = 1, 3, 5, \dots$ . Find: Frequency for closed pipes $f$ .

$$f = \frac{nv}{4L}$$

Given: Slit width $a$ , order of minimum $n$ , and wavelength $\lambda$ . Find: Angle to the dark fringe in single-slit diffraction $\theta$ .

$$a \sin(\theta) = n\lambda$$

Given: Distance between slits $d$ , order $n$ , and wavelength $\lambda$ . Find: Angle to the dark fringe in double-slit interference $\theta$ .

$$d \sin(\theta) = \left(n + \frac{1}{2}\right)\lambda$$

Given: Index of refraction of the first medium $n_1$ and second medium $n_2$ . Find: Critical angle for total internal reflection $\theta_c$ .

$$\theta_c = \sin^{-1}\left(\frac{n_2}{n_1}\right)$$

Given: Index of refraction $n$ and radii of curvature $R_1, R_2$ . Find: Focal length using the Lensmaker's equation $f$ .

$$\frac{1}{f} = (n-1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right)$$

Given: Focal lengths of individual lenses in contact $f_1, f_2, \dots$ . Find: Equivalent focal length $f_{eq}$ .

$$\frac{1}{f_{eq}} = \frac{1}{f_1} + \frac{1}{f_2} + \dots$$

Given: Powers of individual lenses in contact $P_1, P_2, \dots$ . Find: Equivalent power $P_{eq}$ .

$$P_{eq} = P_1 + P_2 + \dots$$

Given: Magnifications of individual lenses $m_1, m_2, \dots$ . Find: Total magnification $M_{total}$ .

$$M_{total} = m_1 \times m_2 \times \dots$$

Given: Mass $m$ and the speed of light $c$ . Find: Equivalent rest energy $E$ .

$$E = mc^2$$

Given: A parent nucleus $^{A}_{Z}X$ . Find: The products of beta-positive (positron) decay.

$$^{A}_{Z}X \rightarrow ^{A}_{Z-1}Y + e^+ + \nu_e$$

Given: An excited parent nucleus $^{A}_{Z}X^*$ . Find: The products of gamma decay.

$$^{A}_{Z}X^* \rightarrow ^{A}_{Z}X + \gamma$$

Given: A parent nucleus $^{A}_{Z}X$ absorbing an inner-shell electron $e^-$ . Find: The products of electron capture.

$$^{A}_{Z}X + e^- \rightarrow ^{A}_{Z-1}Y + \nu_e$$

Given: Decay constant $\lambda$ and number of undecayed nuclei $N$ . Find: The rate of nuclear decay $\frac{\Delta N}{\Delta t}$ .

$$\frac{\Delta N}{\Delta t} = -\lambda N$$

Given: Initial velocity ( $v_0$ ), acceleration ( $a$ ), and time ( $t$ ). Find: Final velocity ( $v_f$ ) without knowing displacement.

$$v_f = v_0 + at$$

Given: Initial velocity ( $v_0$ ), acceleration ( $a$ ), and displacement ( $\Delta x$ ). Find: Final velocity ( $v_f$ ) without knowing time.

$$v_{f}^{}=\sqrt{v_0^2+2a\Delta x}$$

Given: Initial velocity ( $v_0$ ), acceleration ( $a$ ), and time ( $t$ ). Find: Displacement ( $\Delta x$ ).

$$\Delta x = v_0t + \frac{1}{2}at^2$$

Given: An object is launched at an angle $\theta$ with initial velocity $v$ . Find: The vertical and horizontal components of velocity.

Vertical: $v_y = v\sin\theta$ ; Horizontal: $v_x = v\cos\theta$

Given: The mass of an object and its acceleration. Find: The net force acting on the object.

Newton's Second Law: $F_{net} = ma$

Given: The coefficient of static friction ( $\mu_s$ ) and the normal force ( $F_N$ ). Find: The maximum force that can be applied before the object begins to move.

Maximum static friction: $f_{s, max} = \mu_s F_N$

Given: Mass of an object ( $m_1$ ), mass of a planet ( $m_2$ ), and the distance between their centers ( $r$ ). Find: The gravitational force of attraction.

Newton's Law of Universal Gravitation: $F_g = \frac{Gm_1m_2}{r^2}$

Given: Mass ( $m$ ), velocity ( $v$ ), and radius of a circular path ( $r$ ). Find: The force keeping the object in uniform circular motion.

Centripetal Force: $F_c = \frac{mv^2}{r}$

Given: An object of mass $m$ on an inclined plane with angle $\theta$. Find: The components of gravity parallel and perpendicular to the plane.

Parallel to plane: $F_{g\parallel} = mg\sin\theta$ ; Perpendicular to plane (Normal Force if flat): $F_{g\perp} = mg\cos\theta$

Given: Force applied ( $F$ ), distance traveled ( $d$ ), and the angle between the force and displacement vectors ( $\theta$ ). Find: Work done.

$$W = Fd\cos\theta$$

Given: The mass of an object (m) and its velocity (v). Find: The kinetic energy associated with its motion (KE).

Kinetic Energy: $KE = \frac{1}{2}mv^2$

Given: Mass ( $m$ ), height relative to a datum ( $h$ ), and gravity ( $g$ ). Find: Energy associated with a objects position.

Gravitational Potential Energy: $U = mgh$

Given: The spring constant ( $k$ ) and the displacement from equilibrium ( $x$ ). Find: The energy stored in the spring.

Elastic Potential Energy: $U = \frac{1}{2}kx^2$

Given: The net work done on an object. Find: The change in the object's speed/kinetic energy.

Work-Energy Theorem: $W_{net} = \Delta KE = KE_f - KE_i$

Given: Work done ( $W$ ) and the time it took to do it ( $t$ ), OR Force ( $F$ ) and velocity ( $v$ ). Find: Power output.

$$P = \frac{W}{t} = Fv\cos\theta$$

Given: Heat added to a system ( $Q$ ) and work done by the system ( $W$ ). Find: The change in internal energy ( $\Delta U$ ).

First Law of Thermodynamics: $\Delta U = Q - W$

Given: Mass of a substance ( $m$ ), its specific heat capacity ( $c$ ), and a change in temperature ( $\Delta T$ ). Find: Heat gained or lost ( $q$ ) without a phase change.

$$q = mc\Delta T$$

Given: Mass of a substance ( $m$ ) and its latent heat of fusion or vaporization ( $L$ ). Find: Heat required to change its phase.

Phase change heat: $q = mL$ (Remember: Temperature does not change during a phase change).

Relationship: If a gas expands and does work on its surroundings, what happens to the sign of Work ( $W$ ) in the First Law equation?

Work done by the system is positive ( $+W$ ). Expansion = positive work. Compression = negative work.

Given: The density of a fluid ( $\rho$ ), gravity ( $g$ ), and depth below the surface ( $h$ ). Find: The absolute pressure at that depth. (Hint: You need to know atmospheric pressure)

Hydrostatic Pressure: $P = P_0 + \rho gh$ (where $P_0$ is surface/atmospheric pressure).
Atmospheric pressure = 1 atm = 760 mmHg = 760 torr = 1 × 10^5 Pa

Given: The density of a fluid ( $\rho_{fluid}$ ), and the volume of the object submerged ( $V_{sub}$ ). Find: The upward force on the object.

Buoyant Force (Archimedes' Principle): $F_B = \rho_{fluid} V_{sub} g$

Given: Cross-sectional area of a pipe ( $A_1$ ) and fluid velocity ( $v_1$ ) at point 1, and the area ( $A_2$ ) at point 2. Find: Velocity at point 2.

Continuity Equation: $A_1v_1 = A_2v_2$

Given: Pressures, heights, and fluid velocities at two different points in a closed pipe system. Find: A missing pressure or velocity.
(Hint: You must know the density of the liquid)

Bernoulli's Equation: $P_1 + \frac{1}{2}\rho v_1^2 + \rho gh_1 = P_2 + \frac{1}{2}\rho v_2^2 + \rho gh_2$

Density ($\rho$) = $\rho_{w}\cdot SG$ (SG: Specific Gravity Ratio)

Density of Water ($\rho_{w}$ ): 1000 kg/m³ or 1 kg/L

Relationship: Fluid flows through a rigid pipe. If the radius of the pipe is reduced by half, what happens to the flow rate ( $Q$ ) assuming constant pressure difference?

Flow rate decreases by a factor of 16. (Poiseuille's Law: $Q \propto r^4$ ).

Given: The magnitude of two point charges ( $q_1, q_2$ ) and the distance between them ( $r$ ). Find: The electrostatic force between them.
(Hint: You need to know Coulomb’s Constant (k) )

Coulomb's Law: $F_e = \frac{k\vert{}q_1q_2\vert{}}{r^2}$

Coulomb’s Constant (k) = 9 × 10⁹ N * m² / C²

Given: A source charge ( $Q$ ) and the distance from it ( $r$ ). Find: The magnitude of the electric field created by the charge. (Hint: You need Coulomb’s Constant)

Electric Field: $E = \frac{k\vert{}Q\vert{}}{r^2}$

Coulomb’s Constant (k) = 9 × 10⁹ N * m² / C²

Given: The electric field strength ( $E$ ) and a test charge ( $q$ ) placed within it. Find: The force acting on the test charge.

$$F_e = qE$$

Given: A source charge ( $Q$ ) and distance ( $r$ ). Find: The electrical potential (Voltage) at that point in space. (Hint: you need coulomb’s constant)

Electrical Potential: $V = \frac{kQ}{r}$

Coulomb’s constant (k) = 9 × 10⁹ N * m² / C²

Given: A charge ( $q$ ) and the electrical potential/voltage ( $V$ ) it experiences. Find: The electrical potential energy.

$$U = qV$$

Given: The current ( $I$ ) flowing through a resistor and its resistance ( $R$ ). Find: The voltage drop across it.

Ohm's Law: $V = IR$

Given: The resistivity of a material ( $\rho$ ), its length ( $L$ ), and cross-sectional area ( $A$ ). Find: Its resistance.

$$R = \frac{\rho L}{A}$$

Given: Voltage ( $V$ ) and Current ( $I$ ), OR Current and Resistance ( $R$ ). Find: The power dissipated by a resistor.

$$P = IV = I^2R = \frac{V^2}{R}$$

Relationship: You have three resistors ( $R_1, R_2, R_3$ ). How do you calculate total equivalent resistance if they are in Series vs. Parallel?

Series: $R_{eq} = R_1 + R_2 + R_3$ ; Parallel: $\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}$

Given: The charge stored on a capacitor ( $Q$ ) and the voltage across it ( $V$ ). Find: Capacitance.

$$C = \frac{Q}{V}$$

Given: The area of capacitor plates ( $A$ ), distance between them ( $d$ ), the permittivity of free space ($\epsilon_0$), and dielectric constant ( $\kappa$ ). Find: Capacitance (C).

$$C = \frac{\kappa \epsilon_0 A}{d}$$

Given: Capacitance ( $C$ ) and Voltage ( $V$ ). Find: Energy stored in the capacitor.

$$U = \frac{1}{2}CV^2$$

Given: The frequency ( $f$ ) and wavelength ( $\lambda$ ) of a wave. Find: Wave propagation speed.

$$v = f\lambda$$

Relationship: What is the relationship between the period of a wave ( $T$ ) and its frequency ( $f$ )?

Inverse: $f = \frac{1}{T}$ or $T = \frac{1}{f}$

Given: The velocity of a sound source ( $v_s$ ), the velocity of the observer ( $v_o$ ), and the speed of sound ( $v$ ). Find: The perceived frequency ( $f'$ ).

Doppler Effect: $f' = f \left(\frac{v \pm v_o}{v \mp v_s}\right)$ (Top signs when moving toward, bottom signs when moving away).

Relationship: A sound source moves away from you. What happens to the perceived distance between wave peaks?

Wavelength increases (sound perceived drops to a lower frequency/pitch).

Given: The intensity of a sound wave ( $I$ ). Find: The sound level ( $\beta$ ) in decibels (dB).

$\beta = 10 \log\left(\frac{I}{I_0}\right)$ (where $I_0 = 10^{-12} W/m^2$ ).

Given: The speed of light in a vacuum ( $c$ ) and the speed of light in a specific medium ( $v$ ). Find: The index of refraction ( $n$ ).

$$n = \frac{c}{v}$$

Given: The index of refraction of medium 1 ( $n_1$ ), medium 2 ( $n_2$ ), and the angle of incidence ( $\theta_1$ ). Find: The angle of refraction ( $\theta_2$ ).

Snell's Law: $n_1\sin\theta_1 = n_2\sin\theta_2$

Given: The distance to the image ( $d_i$ ) and the distance to the object ( $d_o$ ). Find: The focal length of a lens/mirror ( $f$ )

Thin Lens/Mirror Equation: $\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}$

Given: Image distance ( $d_i$ ) and object distance ( $d_o$ ). Find: Magnification ( $m$ ).

$$m = -\frac{d_i}{d_o}$$

Relationship: If the magnification ( $m$ ) of a lens is a negative number, what does that tell you about the image?

The image is inverted and real. (Positive $m$ = upright and virtual).

Given: The focal length of a lens ( $f$ ) in meters. Find: The power of the lens ( $P$ ) in diopters.

$$P = \frac{1}{f}$$

Given: Planck's Constant ($h$) and the frequency of a photon ( $f$ ) or its wavelength ( $\lambda$ ). Find: The energy of the photon.
(Hint: You might need the speed of light)

$E = hf = \frac{hc}{\lambda}$
Speed of light ($c$) = 3 × 10⁸ m/s

Given: The energy of an incident photon ( $E = hf$ ) and the work function of a metal ( $W$ ). Find: The maximum kinetic energy of the ejected electron.

Photoelectric Effect: $K_{max} = hf - W$

Relationship: An atom undergoes Alpha Decay. What happens to its mass number and atomic number?

Mass number decreases by 4. Atomic number decreases by 2. (Ejects a Helium nucleus).

Relationship: An atom undergoes Beta-Minus Decay. What happens to its mass number and atomic number?

Mass number remains unchanged. Atomic number increases by 1. (A neutron is converted into a proton)

Given: Displacement of spring (x) and spring constant (k) Find: Spring Force (F_s)

Hooke’s Law: F_s = $-kx$