Curvature
Curvature of a circle
What is 1/R ?
The simplest example of: v(t) is constant but vec v(t) is changing
What is uniform circular motion ?
Figured out that the path of an orbiting satellite is an ellipse
Who was Johannes Kepler ?
<x'(t), y'(t), z'(t)>
What is the tangent or velocity vector
vec r'(t) ?
How tall is a stack of new $100 bills worth $1 million? (within 1 foot)
What is 43 inches (3.6 feet) ?
N in kappa = N/(1+f'(x)^2)^(3/2)
What is | f''(x) | ?
vec(v(t))/||vec(v(t))|| or vec(r'(t))/||vec(r'(t))||
What is the unit tangent or velocity vector vec T ?
A satellite that remains above a fixed point on the Earth's equator
What is a geosynchronous satellite ?
sqrt(x'(t)^2+y'(t)^2+z'(t)^2
What is v(t), the speed of
vec r(t) ?
How many times can a light beam travel around the Earth in 1 second? (within 3)
What is 7.5 ?
N in kappa = N/||vec r'(t)||^3
What is
|| r'(t) xx r''(t) || ?
vec (T'(t))/||vec(T'(t))||
What is the unit normal vector vec N ?
A consequence is that if you know the semi-major axis of an orbit and its period, you can calculate the mass of the object the satellite is orbiting around
What is Kepler's Third Law, or the Law of the Period ?
int_a^b sqrt(x'(t)^2+y'(t)^2+z'(t)^2) dt
What is the path length travelled by
vec r(t) ?
How wide are the transistors in the central processing unit (CPU) of a modern PC? (within an order of magnitude)
What is "a few nanometers" ? (1 nanometer = 1 billionth of a meter)
vec r = vec r(t0) + 1/kappa vec N+ R<cos t, sin t>
What is the equation of an osculating circle ?
vec a cdot vec T
What is the tangential component of acceleration?
A consequence is that a planet travels faster at perihelion than at aphelion
What is Kepler's Second Law, the Law of Equal Area in Equal Time ?
A parametrization with ||vec(r'(t))|| = 1
What is an arc length parametrization ?
What is the number of people who may be displaced and buildings made uninhabitable in Florida by 2100, due to sea level rise? (within an order of magnitude)
What is 1 million?
vec B cdot <x-x0, y-y0, z-z0> = 0
What is the equation of an osculating plane ?
kappa(t) v(t)^2
What is the normal component of acceleration ?
N in T^2 = N a^3
What is
(4pi^2)/(GM) ?
Q in Q=vec T xx vec N
What is the binormal vector
vec B ?
What is the typical diameter of an atomic nucleus? (within an order of magnitude)
What is 10^-14 - 10^-15 m ?