x̄ - > Centripetal acceleration | gravitational acceleration

centripetal acceleration 

| tangential speed | 30 mph (miles per hour) 

radius | 500 feet

centripetal acceleration | 118 cm/s^2 (centimeters per second squared)

= 3.872 ft/s^2 (feet per second squared) 

= 9504 mi/h^2 (miles per hour squared) 

= 1.18 m/s^2 (meters per second squared)

a = v^2/r | 

a | centripetal acceleration 

v | tangential speed 

r | radius

Physics jokes A neutron walks into a bar and inquires how much a drink costs. The bartender replies, "For you? No charge." ,

 gravitational acceleration 

g = (G M)/r^2 | 

 g | gravitational acceleration 

M | mass 

r | radius 

G | Newtonian gravitational constant (≈ 6.674×10^-11 m^3/(kg s^2))

mass | Earth (planet) (mass): 5.97×10^24 kg (kilograms) 

radius | Earth (planet) (average radius): 6371.009 km (kilometers)

gravitational acceleration | 9.82 m/s^2 (meters per second squared) 

= 32.22 ft/s^2 (feet per second squared) 

= 982 cm/s^2 (centimeters per second squared) escape 

velocity | 11.19 km/s (kilometers per second) 

11190 m/s (meters per second) 

25020 mph (miles per hour)

damped harmonic oscillator (physical system)

Degrees of freedom 1 

S(t_1, t_2) = integral_(t_1)^(t_2) exp(2 ζ ω_0 τ) (1/2 m x'(τ)^2 - 1/2 m ω_0^2 x(τ)^2) dτ 

 

L = exp(2 ζ ω_0 t) (1/2 m x'(t)^2 - 1/2 m ω_0^2 x(t)^2) 

 

x(t) = e^(ζ (-t) ω_0) ((x'(0) sinh(α t))/α + x(0) ((ζ ω_0 sinh(α t))/α + cosh(α t))) 

with α = sqrt(-1 + ζ^2) ω_0 

 

ℋ = exp(-2 ζ ω_0 t) p(t)^2/(2 m) + exp(2 ζ ω_0 t) 1/2 m ω_0^2 x(t)^2 

 x'(t) = exp(-2 ζ ω_0 t) p(t)/m | 

p'(t) = -exp(2 ζ ω_0 t) m ω_0^2 x(t) 

 

x(t) = e^(ζ (-t) ω_0) ((p(0) sinh(α t))/(α m) + x(0) ((ζ ω_0 sinh(α t))/α + cosh(α t))) 

with α = sqrt(-1 + ζ^2) ω_0 p(t) = e^(ζ t ω_0) (p(0) (cosh(α t) - (ζ ω_0 sinh(α t))/α) - (m x(0) ω_0^2 sinh(α t))/α) with α = sqrt(-1 + ζ^2) ω_0 

 p(t) = m e^(2 ζ t ω_0) x'(t) T = (2 π)/(sqrt(1 - ζ^2) ω_0) for abs(ζ)