APPC Equations

Acceleration due to gravity at a certain altitude

g = GM/(r+h)²

Acceleration due to gravity given planet radius and mass

g = GM/(r²)

Acceleration due to gravity at a certain altitude

g = GM/(r+h)²

Acceleration due to gravity given planet radius and mass

g = GM/(r²)

Angular frequency given the actual frequency

ω =2πf

Angular momentum for a particle (individual)

L = r x p

must be CROSS PRODUCT

Angular momentum for a rigid object

L = Iω

Average acceleration

a = ∆v/∆t

Average angular acceleration

α = Δω/Δt

Centripetal acceleration

a = v²/r

Conservation of angular momentum

L(final) = L(initial)

Conservation of mechanical energy (w/o friction)

∑E(final) = ∑E(initial)

Conservation of mechanical energy (with friction)

∑E(final) ≠ ∑E(initial)

Conservation of Momentum - Elastic Collisions

p(final) = p(initial)

KE is conserved

Conservation of Momentum - Inelastic Collisions

p(final) = p(initial)

KE is NOT conserved

Constant freefall acceleration given initial and final vertical velocities and a change in vertical position

(vf)² = (vi)² + 2g(∆y)

Constant freefall acceleration given initial and final vertical velocities and time

∆v = gt

Constant freefall acceleration given initial vertical velocity, a change in vertical position, and time

∆y = (vi)t + 0.5gt²

Constant horizontal acceleration given initial and final velocities and a change in horizontal position

(vi)² = (vf)² + 2a(∆x)

Constant horizontal acceleration given initial and final velocities and time

∆v = at

Constant horizontal acceleration given initial velocity, a change in horizontal position, and time

∆x = (vi)t + 0.5at²

Direction of resultant vector

Trig :)

Displacement

∆x = (vi)t + 0.5at² = 0.5(∆v)t

Elastic Potential energy or work done on a spring

U = 0.5k(∆x)²

Equation for distance (position) of an object due to the velocity, acceleration, and time

∆x = (vi)t + 0.5at²

Equation for the final angular velocity due to an initial velocity, acceleration, & angular position

(ωf)² = (ωi)² + 2⍺(∆θ)

Equation for the final velocity due to an initial velocity, acceleration, and time

vf = vi + at

Escape velocity given "planet's" mass and radius

v = √((2GM)/(r))

Frequency given the period

f = 1/T

Gravitational Potential Energy

U = mg∆h

Gravitational Potential energy given two masses and the distance between them

U = -(GMm)/r

Horizontal distance of a projectile

x=(vi)tcos(θ)

Impulse due to a force applied over a small time or due to a change in momentum

j = ∫F dt = ∆p

Impulse due to a varying force

j = ∫F dt

Instantaneous acceleration

a(t) = v'(t)

Instantaneous angular acceleration

α(t) = ω'(t)

Instantaneous angular velocity

ω(t) = θ'(t)

Instantaneous angular velocity due to the integration of the angular acceleration function

v(t) = ∫a(t) dt

Instantaneous Power

P = ∑E'(t)

Instantaneous velocity

v(t) = x'(t)

Kinetic energy

K = 1/2 mv²

Kinetic Friction

Ff = µFn

Linear Momentum

p = mv

Magnitude of resultant vector

a² + b² = c²

Maximum acceleration of an oscillating object given the angular frequency

a = Aω²

Maximum acceleration of an oscillating object given the spring constant

a = (k/m)A

Mechanical Power

P = Fv = W/t

Moment of inertia (rotational inertia) for solid disk

I = 0.5MR²

Moment of inertia (rotational inertia) for a particle

I = MR²

Net Centripetal force

F = m a© = mv²/r

Net force resulting in acceleration

Fnet = ma

Net torque is proportional to its angular acceleration

τ = I⍺

Net work due to a change in kinetic energy

W = ∆KE

Newton's Law of Universal Gravitation (Gravitation Force)

F = (GMm)/r²

Parallel-axis theorem

I = I(cm) + mh²

Power delivered to a rotating rigid object

P = τω

Range of a projectile given the initial velocity and angle of elevation

R = v² sin(2θ) / g

Relation of force between members of a system to the potential energy of the system

F = -dU/dx

Relationship between arc length and angle

arclength = 2πr(θ/360)

Relationship between tangential (linear) acceleration and angular acceleration

a = r⍺

Relationship between tangential (linear) velocity and angular velocity

v = rω

Right triangle trigonometric ratios

sin = 0, 1, 2, 3, 4
cos = 4, 3, 2, 1, 0
tan = sin/cos

Rotational equilibrium-clockwise and counterclockwise torques are balanced

τ(⤴) = τ(⤵)

Rotational kinetic energy

K = 1/2 Iω²

Spring force

F = -k∆x

Static Friction

F(static friction) = µₛFₙ

The torque due to a component of force perpendicular to a lever

τ = rFsin(θ)

Time period given the angular frequency

T = 2π/ω

Time period of a mass oscillating on a spring

T = 2π√(m/k)

Time period of a pendulum of length L

T = 2π√(L/g)

Total acceleration due to an objects centripetal and tangential acceleration in non-uniform circular motion

aₜₒₜ = √(a꜀² + aₜ²)

Total angular displacement due to the integration of the angular velocity function

∆θ = ∫ω dt

Total energy for circular orbits

E = K + U = -(GMm)/(2r)

Total energy for elliptical orbits

E = K + U = -(GMm)/(R + r)

Total energy of an orbiting body

E = -(GMm)/(2a)

Total kinetic energy of a rolling object

KE = 0.5m(v꜀ₘ)² + 0.5I꜀ₘ(ω)²

Total mechanical energy

E = KE + U

Two formulas for average angular velocity

ωₐᵥ₉ = ∆θ/∆t = vₐᵥ₉r

Velocity of an object in uniform circular motion

v = √(a꜀r)

Velocity of an object oscillating in simple harmonic motion

v(t) = -vₘₐₓ sin(ωt + ɸ)

Weight

W = F₉ = mg

Work due to a constant force

W = ∫F dx

Work-kinetic energy theorem for rotation

W = ∫τ dθ

X and Y components for an object's center of mass

Xcm = (∑(mx))/(∑m)