List of Physics Equations

In learning Physics, there will be many equations that one will use repeatedly (and should memorize). This is my attempt to pool every Physics equation I come across in my notes/studies.

Note that there may be some overlap between this page and other topic equation pages, such as Mathematics itself (displacement, velocity, and acceleration, for example) or Electrical Engineering, for topics like Electromagnetism. All equations are inherently mathematical, but many are only ever used in a math-applied field for a specified purpose, such as how Tsiolkovsky's rocket equation will be taught in Astronautic engineering class, but not in a higher-level calculus class. Equations used in mathematics are found here, while those used in electrical engineering are found here.

In the unlikely event there are any egregious errors in this page, contact me at jdlacabe@gmail.com.

I.I Basics of Mechanics.

DENSITY:

d = (m / V)

d = Density
m = Mass
V = Volume

I.II Accuracy & Uncertainty.

RELATIVE ERROR:

Er = ((O - A) / A) × 100

A = Accepted Value
Er = Relative Error
O = Observed Error

I.III Displacement and V.S.A.

DISPLACEMENT (IN UNIT VECTORS):

d = ((x₂ - x₁)î + (y₂ - y₁)ĵ)

[IN 3D]:

d = ((x₂ - x₁)î + (y₂ - y₁)ĵ + (z₂ - z₁)k̂)

d = Displacement
x₁ = The x-component of the first vector.
x₂ = The x-component of the second vector.
y₁ = The y-component of the first vector.
y₂ = The y-component of the second vector.
z₁ = The z-component of the first vector.
z₂ = The z-component of the second vector.
î = The unit vector indicating direction along the x-axis.
ĵ = The unit vector indicating direction along the y-axis.

DISPLACEMENT (THE MAGNITUDE THEREOF):

|d| = √((x₂ - x₁)² + (y₂ - y₁)²)

[IN 3D]:

|d| = √((x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²)

d = Displacement
x₁ = The x-component of the first vector.
x₂ = The x-component of the second vector.
y₁ = The y-component of the first vector.
y₂ = The y-component of the second vector.
z₁ = The z-component of the first vector.
z₂ = The z-component of the second vector.

AVERAGE VELOCITY:

V_avg = (d / ∆t) = (x_F - x_i) / (t_F - t_i)

V_avg = Average Velocity
d = Displacement
x_i = The initial instantaneous position
x_F = The final instantaneous position
∆t = Change in time
t_i = The initial time value
t_F = The final time value

INSTANTANEOUS VELOCITY:

dr/dt = v_{instantaneous}

dr = Instantaneous position of the Vector
dt = Instantaneous time (an exact time)
v_{instantaneous} = Instantaneous Velocity

AVERAGE ACCELERATION:

a_avg = (∆v / ∆t) = (v_F - v_i) / (t_F - t_i)

a_avg = Average Acceleration
∆v = Change in velocity
v_i = The initial instantaneous velocity
v_F = The final instantaneous velocity
∆t = Change in time
t_i = The initial time value
t_F = The final time value

INSTANTANEOUS ACCELERATION:

dv/dt = a_{instantaneous}

dv = Instantaneous Velocity of the Vector
dt = Instantaneous time (an exact time)
a_{instantaneous} = Instantaneous Acceleration

SPEED:

s = (total distance) / (∆t)

s = Speed
∆t = Change in time

I.IV Uniformly Accelerated Motion.

UNIFORMLY ACCELERATED MOTION:

V_F = V_i + (a × ∆t).
∆x = (V_i × ∆t) + (1/2 × a × ∆t²)
V_F² = V_i² + (2 × a × ∆x)
∆x = (1/2) × (V_F + V_i) × ∆t

V_F = Final Instantaneous Velocity
V_i = Initial Instantaneous Velocity
a = Average Acceleration (really!)
∆t = Change in time
∆x = Displacement (or change in position)

III.I Vector Basics.

VECTOR SUBTRACTION:

A - B = A + (-B)

A & B = Any Two Vectors

ASSOCIATIVE PROPERTY OF VECTOR ADDITION:

A + (B + C) = (A + B) + C

A & B & C = Any Two Vectors

3D VECTOR REPRESENTED IN UNIT VECTORS:

r = xî + yĵ + zk̂

r = Any Vector
xî = The Vector's x-direction component
yĵ = The Vector's y-direction component
zk̂ = The Vector's z-direction component

III.III Rotation & Multiplication.

VECTOR ROTATION:

θ = θ′ + ϕ

θ = The original angle between the vector and the x-axis pre-rotation.
θ′ = The angle between the vector and the new 'x'-axis post-rotation.
ϕ = The angular distance between the original x-xis and the rotated x-axis.

DOT PRODUCT IN 2D:

a · b = abcosϕ

No direction - it is scalar.

a = The magnitude of a
b = The magnitude of b
ϕ = The angle between a and b

DOT PRODUCT IN 3D:

a · b = aₓbₓ + aᵧbᵧ + a_zb_z

a & b = Any Two Vectors

CROSS PRODUCT IN 2D:

a × b = absinϕ
The direction is found using the right-hand rule.

a = The magnitude of a
b = The magnitude of b
ϕ = The angle between a and b

CROSS PRODUCT IN 3D:

a × b = (aᵧb_z - bᵧa_z)î + (a_zbₓ - b_zaₓ)ĵ + (aₓbᵧ - bₓaᵧ)k̂

a & b = Any Two Vectors

III.IV Drag.

FORCE OF DRAG:

F_drag = (1/2) × ρ × V² × D × A

ρ (rho) = Density of the medium
V = Velocity of the object
D = Drag Coefficient
A = Cross Sectional Area

IV.I Basics of Range.

RANGE:

Range = (V_i² × sin(2θ_i)) / g

R = Range = Δx
V_i = ||V_i|| (Magnitude of Vi)
θ_i = Initial Angle or Launch Angle
g = Acceleration due to gravity on Earth, POSITIVE 9.81 m/s²

V.I Angular & Tangential Velocity.

Arc Length:

s = r × Δθ

s = Arc Length
r = Radius
Δθ = Angular Displacement

AVERAGE ANGULAR VELOCITY:

ω_avg = (∆θ / ∆t)

ω_avg = Average Angular Velocity
∆θ = Angular Displacement
∆t = Change in time

INSTANTANEOUS ANGULAR VELOCITY:

dθ/dt = ω_{instantaneous}

dθ = Instantaneous Angular Displacement
dt = Instantaneous time (an exact time)
ω_{instantaneous} = Instantaneous Angular Velocity

AVERAGE ANGULAR ACCELERATION:

α_avg = (∆ω / ∆t)

α_avg = Average Angular Acceleration
∆ω = Change in Angular Velocity
∆t = Change in time

INSTANTANEOUS ANGULAR ACCELERATION:

dω/dt = α_{instantaneous}

dv = Instantaneous Angular Velocity
dt = Instantaneous time (an exact time)
α_{instantaneous} = Instantaneous Angular Acceleration

UNIFORMLY ACCELERATED ANGULAR MOTION:

ω_F = ω_i + (α × ∆t).
∆θ = (V_i × ∆t) + (1/2 × α × ∆t²)
ω_F² = ω_i² + (2 × α × ∆θ)
∆θ = (1/2) × (ω_F + ω_i) × ∆t

ω_F = Final Instantaneous Angular Velocity
ω_i = Initial Instantaneous Angular Velocity
α = Average Angular Acceleration
∆t = Change in time
∆θ = Angular Displacement (or change in angular position)

TANGENTIAL VELOCITY:

v_t = r × ω

v_t = Tangential Velocity (average or instantaneous, depending on the angular velocity)
r = Radius
ω = Angular Velocity (average or instantaneous)

TANGENTIAL ACCELERATION:

a_t = r × α

a_t = Tangential Acceleration (average or instantaneous, depending on the angular acceleration)
r = Radius
α = Angular Acceleration (average or instantaneous)

V.II Centripetal Velocity.

CENTRIPETAL ACCELERATION:

a_c = (V_t²) / r

a_c = Centripetal Acceleration
V_t² = Tangential Velocity
r = Radius

Table Of Contents

DENSITY:

RELATIVE ERROR:

DISPLACEMENT (IN UNIT VECTORS):

[IN 3D]:

DISPLACEMENT (THE MAGNITUDE THEREOF):

[IN 3D]:

AVERAGE VELOCITY:

INSTANTANEOUS VELOCITY:

AVERAGE ACCELERATION:

INSTANTANEOUS ACCELERATION:

SPEED:

UNIFORMLY ACCELERATED MOTION:

VECTOR SUBTRACTION:

ASSOCIATIVE PROPERTY OF VECTOR ADDITION:

3D VECTOR REPRESENTED IN UNIT VECTORS:

VECTOR ROTATION:

DOT PRODUCT IN 2D:

DOT PRODUCT IN 3D:

CROSS PRODUCT IN 2D:

CROSS PRODUCT IN 3D:

FORCE OF DRAG:

RANGE:

Arc Length:

AVERAGE ANGULAR VELOCITY:

INSTANTANEOUS ANGULAR VELOCITY:

AVERAGE ANGULAR ACCELERATION:

INSTANTANEOUS ANGULAR ACCELERATION:

UNIFORMLY ACCELERATED ANGULAR MOTION:

TANGENTIAL VELOCITY:

TANGENTIAL ACCELERATION:

CENTRIPETAL ACCELERATION: