Physics Formulas Reference

Some of the most important and frequently used formulas in physics are prsented and explained below.

Kinematics (Quantitative Description of Motion)

Formula	Definition and explanations
\( s_{av} = \dfrac{d}{\Delta t} \)	s_av is the average speed (scalar) d is the distance Δ t is the time elapsed
\( v_{av} = \dfrac{x_f - x_i}{t_f - t_i} =\dfrac{\Delta x}{\Delta t} \)	v_av is the average velocity (vector) Δ x is the displacement(vector) Δ t is the time elapsed
\( a_{av} = \dfrac{v_f - v_i}{t_f - t_i} =\dfrac{\Delta v}{\Delta t} \)	a_av is the average acceleartion (vector) Δ v is the change in velocity (vector) Δ t is the time elapsed
\( v_{av} = \dfrac{v_i + v_f}{2} \)	v_av is the average velocity (vector) v_i is the initial velocity (vector) v_f is the final velocity (vector)
\( v_{f} = v_{i} + a \Delta t \)	v_f is the final velocity (vector) v_i is the initial velocity (vector) a is the acceleration (vector)
\( \Delta x = v_i \Delta t + \dfrac{1}{2} a (\Delta t)^2 \)	Δ x is the displacement (vector) v_i is the initial velocity (vector) a is the acceleration (vector)
\( \Delta x = v_f \Delta t - \dfrac{1}{2} a (\Delta t)^2 \)	Δ x is the displacement (vector) v_f is the final velocity (vector) a is the acceleration (vector)
\( \Delta x = \dfrac{v_f+v_i}{2} \Delta t \)	Δ x is the displacement (vector) v_f is the final velocity (vector) v_i is the initial velocity (vector)
\( v^2_f = v^2_i + 2 a \cdot \Delta x \)	v_f is the final velocity (vector) v_i is the initial velocity (vector) Δ x is the displacement (vector) a is the acceleration (vector)

Relative Velocity

Formula	Definition and explanations
\( v_{AC} = v_{AB}+v_{BC} \)	v_AC is the velocity of A with respect to C (vector) v_AB is the velocity of A with respect to B (vector) v_BC is the velocity of B with respect to C (vector)

Kinematics (Quantitative Description of Projectile Motion)

Example of projectile motion

Formula	Definition and explanations
\( v_{ix} = \|v_i\|\cos(\theta) \\ v_{iy} = \|v_i\|\sin(\theta) \)	v_i is the initial velocity (vector) v_ix is the component of the initial velocity along the horizontal direction x (scalar) v_iy is the component of the initial velocity along the vertical direction y (scalar) θ is the initial angle that v_i makes with the horizontal.
\( \Delta x = \|v_i\|\cos(\theta) \Delta t \)	Δx is the displacement along the horizontal direction x
\( \Delta y = \|v_i\| \sin(\theta) \Delta t - \dfrac{1}{2} g (\Delta t)^2 \)	Δy is the displacement along the vertical direction y
\( R = \dfrac{v^2_i \sin(2\theta)}{g}\)	R is the range or horizontal distance travelled when the projectile hits the ground
\( T = \dfrac{2 v_i \sin(\theta)}{g}\)	T is total time to hit the ground
\( H = \dfrac{v^2_i \sin^2(\theta)}{2 g}\)	H maximum height reached above the ground g = 9.8 m / s²

Dynamics (Forces and Momentum)

Formula	Definition and explanations
\( F = m a \)	F is the net force (vector) m is the mass a is the acceleration (vector)
\( F_g = m g \)	F_g is the weight (vector) m is the mass g is the acceleration (near the Earth) due to gravitation (vector)
\( \| F_f \| = \mu \| F_N \| \)	F_f is the force of friction (vector) μ is the coefficient of friction (μ may be μ_k kinetic coefficient or μ_s static coefficient of friction) F_N is the normal (to the surface) force (vector)
\( p = m v \)	p is the momentum (vector) m is the mass v is the velocity (vector)
\( \Delta p = F \Delta t \)	Δ p is the change in momentum (vector) F is the applied force (vector) Δ t is the elapsed time (F Δ t) is called impulse (vector)

Circular Motion

Formula	Definition and explanations
\( a_c = \dfrac{v^2}{r} \)	a_c is the centripetal acceleration v is the velocity r is the radius
\( F_c = \dfrac{m v^2}{r} \)	F_c is the centripetal force v is the velocity m is the mass r is the radius
\( v = \dfrac{2 \pi r}{T} \)	v is the velocity r is the radius T is the period (time for one complete revolution)

Work, Potential and Kinetic Energies

Formula	Definition and explanations
\( W = F d \cos(\theta) \)	W is the work done by the force F F is the applied force (constant) d is the distance θ is the angle between F and the direction of motion
\( E_k = \dfrac{1}{2} m v^2 \)	E_k is the kinetic energy v is the velocity m is the mass
\( E_p = m g h \)	E_p is the potential energy of an object close to the surface of Earth m is the mass of the object h is the height of the object with respect to some refernce (ground for example) g = 9.8 m/s²
\( E_t = E_k + E_p \)	E_t is the total energy E_k is the kinetic energy E_p is the potential energy

Springs, Hooke's Law and Potential Energy

Formula	Definition and explanations
\( F_s = k x \)	F is the force applied to compress or stretch a spring k is the spring constant x is the length of extension or compression of the spring
\( E_s = \dfrac{1}{2} k x^2 \)	E_s is the potential energy stored in a spring when compressed or extended k is the spring constant x is the length of extension or compression of the spring

Period of Simple Harmonic Motions

Formula	Definition and explanations
\( T_s = 2\pi \sqrt{\dfrac{m}{k}} \)	T_s is the time period of motion k is the spring constant m is the mass attached to the spring
\( T_p = 2\pi \sqrt{\dfrac{L}{g}} \)	E_p is the time period of motion L is the length of the pendilum g is the acceleration due to gravity

Gravitational Fields and Forces

Formula	Definition and explanations
\( F = G \dfrac{m_1 m_2}{r^2} \)	F is force of attraction G is the universal gravitational constant m₁ and m₁ are the masses of the two objects attracting each other r is the distance separating the centers of the two objects
\( g_r = \dfrac{G m}{r^2} \)	g_r gravitational field intensity at a distance r G is the universal gravitational constant m is the mass r is the distance (from mass m) where the field is measured
\( E_p = -\dfrac{G M m}{r} \)	E_p gravitational potential energy of mass m G is the universal gravitational constant G is the mass of the attracting body m is the mass being attracted r is the distance separating the centers of the masses M and m

Satelite motion, orbital speed, period and radius

Formula	Definition and explanations
\( v = \sqrt{ \dfrac{G M}{r} } \)	v is the orbital speed of the satellite G is the universal gravitational constant M is the mass of the attracting body (Earth for example) r is the distance from the center of mass M to the position of the satellite
\( T = \sqrt{ \dfrac{4\pi^2r^3}{G M} } \)	T is the orbital period of the satellite G is the universal gravitational constant m is the mass r is the distance from the center of mass M to the the position of the satellite
\( v = \dfrac{2\pi r }{T} \)	v is the orbital speed of the satellite r is the distance from the center of mass M to the the position of the satellite T is the orbital period of the satellite

Electric forces, fields and potentials

Formula	Definition and explanations
\( F = k \dfrac{q_1 q_2}{r^2} \)	F is the electric force k is a constant q₁ and q₁ are the charges attracting or repulsing each other r is the distance separating the two charges
\( F = q E \)	F is the electric force q is the charge E is the eletcric field
\( E = k \dfrac{q}{r^2} \)	E is the electric field due charge q k is a constant q is the charge r is the distance from the charge q where E is being calculated
\( E_p = k \dfrac{q_1 q_2}{r} \)	E_p is the electric potential energy for a system of two charges k is a constant q₁ and q₁ are the charges r is the distance separating the two charges
\( V = k \dfrac{q}{r} \)	V is the electric potential k is a constant q is the charge r is the distance from the charge q
\( E = \dfrac{V}{d} \)	E is the electric field between two large, oppositely charged, conducting parallel plates V is the electric potential difference between the plates d is the distance separating the two plates

Magnetic fields and forces

Formula	Definition and explanations
\( B = \dfrac{\mu _0 I}{2 \pi r} \)	B is magnetic field due to current I in a long conductor of length L μ₀ is permeability in vacuum I the current in the conductor L is the length of the conductor r is the distance from the conductor to where the field B is calculated
\( B = \dfrac{\mu _0 N I}{L} \)	B is the magnetic field (in the center of the solenoid) due to current I in a solenoid of length L μ₀ is permeability in vacuum I the current in the solenoid L is the length of the solenoid N is the number of turns of the solenoid
\( F_m = q v B \sin(\theta) \)	F_m is the magnetic force (due to B) on a charge q moving at a velocity v B the magnetic field θ is the angle between B and the direction of motion of q
\( F_m = I L B \sin(\theta) \)	F_m is the magnetic force (due to B) on a wire with current I and length L B the magnetic field θ is the angle between B and the wire
\( F_m = \dfrac{ \mu _0 I_1 I_2 L }{2 \pi r} \)	F_m is the magnetic force of attraction or repulsion between two parallel wires μ₀ is permeability in vacuum I₁ and I₂ are the currents in the two wires L is the common length between the two wires

Waves

Formula	Definition and explanations
\( v = \lambda f \)	v is the wave velocity λ is the wavelength f is the frequency
\( f = \dfrac{1}{T} \)	f is the wave frequency T is the period of the wave

Optics

Formula	Definition and explanations
\( v = \dfrac{c}{n} \)	v is the velocity of light in a medium of index n c is speed of light in vacuum ( = 3.0 � 10⁸m/s) n is the index of refraction of the medium
\( n_1 \sin \theta_1 = n_2 \sin \theta_2 \)	n₁ is the index of refraction of medium 1 n₂ is the index of refraction of medium 2 θ₁ is the angle of incidence in medium 1 θ₂ is the angle of refraction in medium 2
\( \theta_c = \sin^{-1}(\dfrac{n_2}{n_1}) \)	θ_c is the critical angle such that when the angle of incidence is bigger that θ_c all light is reflected to medium 1 n₁ is the index of refraction of medium 1 (medium of incidence) n₂ is the index of refraction of medium 2 (medium of refraction)
\( \dfrac{1}{D_0} + \dfrac{1}{D_i} = \dfrac{1}{F} \)	D₀ is the distance to the object D_i is the distance to the image F is the focal length

Photoelectric Effects

Formula	Definition and explanations
\( E = h f \)	E is the energy of the photon h is Plank's constant f is the wave frequency of the photon
\( E_k = h f - \phi \)	E_k is the kinetic energy h is Plank's constant f is the wave frequency of the photon φ is the work function of the metal (minimum work required to extract an electron)
\( p = \dfrac{h}{\lambda} \)	p is the momentum of the photon h is Plank's constant λ is the photon wavelength

DC Circuits

Formula	Definition and explanations
\( V = R I \)	V is the voltage across a resistor R is the resistance of the resistor I is the current through the resistor
\( P = I^2 R = \dfrac{V^2}{R} = I V \)	P is the power dissipated as heat into a resistor I is current through the resistor R is the resistance of the resistor V is the voltage across the resistor
\( R_s = R_1 + R_2+... \)	R_s is the total resistance equivalent to several resistors in series (end to end) R₁ resistance of resistor 1 R₂ resistance of resistor 2
\( \dfrac{1}{R_p} = \dfrac{1}{R_1} + \dfrac{1}{R_2} ... \)	R_p is the total resistance equivalent to several resistors in parallel (side by side) R₁ resistance of resistor 1 R₂ resistance of resistor 2
\( C = \dfrac{\epsilon A}{d} \)	C is the capacitance of a capacitor made up of two parallel plates ε is the permittivity of the dielectric inside the two plates A is the common area of the two plates d is the distance between the two plates
\( Q = C V \)	Q is the total charge in a capacitor made up of two parallel plates C is the capacitance V is the voltage across the capacitor
\( W = \dfrac{C V^2}{2} \)	W is the total energy stored in a capacitor C is the capacitance V is the voltage across the capacitor

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