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.
List of Physics Equations
Table Of Contents
Classical Mechanics
Kinematics:
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:
Vavg = (d / ∆t) = (xF - xi) / (tF - ti)
Vavg = Average Velocity
d = Displacement
xi = The initial instantaneous position
xF = The final instantaneous position
∆t = Change in time
ti = The initial time value
tF = The final time value
INSTANTANEOUS VELOCITY:
dr/dt = vinstantaneous
dr = Instantaneous position of the Vector
dt = Instantaneous time (an exact time)
vinstantaneous = Instantaneous Velocity
AVERAGE ACCELERATION:
aavg = (∆v / ∆t) = (vF - vi) / (tF - ti)
aavg = Average Acceleration
∆v = Change in velocity
vi = The initial instantaneous velocity
vF = The final instantaneous velocity
∆t = Change in time
ti = The initial time value
tF = The final time value
INSTANTANEOUS ACCELERATION:
dv/dt = ainstantaneous
dv = Instantaneous Velocity of the Vector
dt = Instantaneous time (an exact time)
ainstantaneous = Instantaneous Acceleration
SPEED:
s = (total distance) / (∆t)
s = Speed
∆t = Change in time
I.IV Uniformly Accelerated Motion.
UNIFORMLY ACCELERATED MOTION:
VF = Vi + (a × ∆t).
∆x = (Vi × ∆t) + (1/2 × a × ∆t²)
VF² = Vi² + (2 × a × ∆x)
∆x = (1/2) × (VF + Vi) × ∆t
VF = Final Instantaneous Velocity
Vi = 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 + azbz
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 = (aᵧbz - bᵧaz)î + (azbₓ - bzaₓ)ĵ + (aₓbᵧ - bₓaᵧ)k̂
a & b = Any Two Vectors
III.IV Drag.
FORCE OF DRAG:
Fdrag = (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 = (Vi² × sin(2θi)) / g
R = Range = Δx
Vi = ||Vi|| (Magnitude of Vi)
θi = 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:
ωavg = (∆θ / ∆t)
ωavg = Average Angular Velocity
∆θ = Angular Displacement
∆t = Change in time
INSTANTANEOUS ANGULAR VELOCITY:
dθ/dt = ωinstantaneous
dθ = Instantaneous Angular Displacement
dt = Instantaneous time (an exact time)
ωinstantaneous = Instantaneous Angular Velocity
AVERAGE ANGULAR ACCELERATION:
αavg = (∆ω / ∆t)
αavg = Average Angular Acceleration
∆ω = Change in Angular Velocity
∆t = Change in time
INSTANTANEOUS ANGULAR ACCELERATION:
dω/dt = αinstantaneous
dv = Instantaneous Angular Velocity
dt = Instantaneous time (an exact time)
αinstantaneous = Instantaneous Angular Acceleration
UNIFORMLY ACCELERATED ANGULAR MOTION:
ωF = ωi + (α × ∆t).
∆θ = (Vi × ∆t) + (1/2 × α × ∆t²)
ωF² = ωi² + (2 × α × ∆θ)
∆θ = (1/2) × (ωF + ωi) × ∆t
ωF = Final Instantaneous Angular Velocity
ωi = Initial Instantaneous Angular Velocity
α = Average Angular Acceleration
∆t = Change in time
∆θ = Angular Displacement (or change in angular position)
TANGENTIAL VELOCITY:
vt = r × ω
vt = Tangential Velocity (average or instantaneous, depending on the angular velocity)
r = Radius
ω = Angular Velocity (average or instantaneous)
TANGENTIAL ACCELERATION:
at = r × α
at = Tangential Acceleration (average or instantaneous, depending on the angular acceleration)
r = Radius
α = Angular Acceleration (average or instantaneous)
V.II Centripetal Velocity.
CENTRIPETAL ACCELERATION:
ac = (Vt²) / r
ac = Centripetal Acceleration
Vt² = Tangential Velocity
r = Radius
