Physics Formulas (Situational)

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.

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

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)