# 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
$$s_{av} = \dfrac{d}{\Delta t}$$ sav is the average speed (scalar)
d is the distance
Δ t is the time elapsed
$$v_{av} = \dfrac{x_f - x_i}{t_f - t_i} =\dfrac{\Delta x}{\Delta t}$$ vav is the average velocity (vector)
Δ x is the displacement(vector)
Δ t is the time elapsed
$$a_{av} = \dfrac{v_f - v_i}{t_f - t_i} =\dfrac{\Delta v}{\Delta t}$$ aav is the average acceleartion (vector)
Δ v is the change in velocity (vector)
Δ t is the time elapsed
$$v_{av} = \dfrac{v_i + v_f}{2}$$ vav is the average velocity (vector)
vi is the initial velocity (vector)
vf is the final velocity (vector)
$$v_{f} = v_{i} + a \Delta t$$ vf is the final velocity (vector)
vi 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)
vi 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)
vf 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)
vf is the final velocity (vector)
vi is the initial velocity (vector)
$$v^2_f = v^2_i + 2 a \cdot \Delta x$$ vf is the final velocity (vector)
vi is the initial velocity (vector)
Δ x is the displacement (vector)
a is the acceleration (vector)

## Relative Velocity

Formula Definition and explanations
$$v_{AC} = v_{AB}+v_{BC}$$ vAC is the velocity of A with respect to C (vector)
vAB is the velocity of A with respect to B (vector)
vBC is the velocity of B with respect to C (vector)

## Kinematics (Quantitative Description of Projectile Motion)

Formula Definition and explanations
$$v_{ix} = |v_i|\cos(\theta) \\ v_{iy} = |v_i|\sin(\theta)$$ vi is the initial velocity (vector)
vix is the component of the initial velocity along the horizontal direction x (scalar)
viy is the component of the initial velocity along the vertical direction y (scalar)
θ is the initial angle that vi 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)
$$F_g = m g$$ Fg is the weight (vector)
m is the mass
g is the acceleration (near the Earth) due to gravitation (vector)
$$| F_f | = \mu | F_N |$$ Ff is the force of friction (vector)
μ is the coefficient of friction (μ may be μk kinetic coefficient or μs static coefficient of friction)
FN 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
$$a_c = \dfrac{v^2}{r}$$ ac is the centripetal acceleration
v is the velocity
$$F_c = \dfrac{m v^2}{r}$$ Fc is the centripetal force
v is the velocity
m is the mass
$$v = \dfrac{2 \pi r}{T}$$ v is the velocity
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
$$E_k = \dfrac{1}{2} m v^2$$ Ek is the kinetic energy
v is the velocity
m is the mass
$$E_p = m g h$$ Ep 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
$$E_t = E_k + E_p$$ Et is the total energy
Ek is the kinetic energy
Ep 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
$$E_s = \dfrac{1}{2} k x^2$$ Es 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
$$T_s = 2\pi \sqrt{\dfrac{m}{k}}$$ Ts is the time period of motion
k is the spring constant
m is the mass attached to the spring
$$T_p = 2\pi \sqrt{\dfrac{L}{g}}$$ Ep 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
$$g_r = \dfrac{G m}{r^2}$$ gr 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
$$E_p = -\dfrac{G M m}{r}$$ Ep 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
$$E_p = k \dfrac{q_1 q_2}{r}$$ Ep 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
$$F_m = q v B \sin(\theta)$$ Fm 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
$$F_m = I L B \sin(\theta)$$ Fm 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
$$F_m = \dfrac{ \mu _0 I_1 I_2 L }{2 \pi r}$$ Fm 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
$$\theta_c = \sin^{-1}(\dfrac{n_2}{n_1})$$ θc is the critical angle such that when the angle of incidence is bigger that θc 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
Di 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
$$E_k = h f - \phi$$ Ek 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
$$R_s = R_1 + R_2+...$$ Rs is the total resistance equivalent to several resistors in series (end to end)
R1 resistance of resistor 1
R2 resistance of resistor 2
$$\dfrac{1}{R_p} = \dfrac{1}{R_1} + \dfrac{1}{R_2} ...$$ Rp 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