Sec1Sess19

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Parametric Equation
Velocity and acceleration

Position vector: r(t) = <x(t),y(t),z(t)>
<math>\vec{r(t)} = \langle t - sin(t), 1 - cos(t) \rangle </math>

the position vector shows y = t*x, so t = y/x and can be substituted into equations
to find xy-equations of a curve

velocity = derivative of position vector with respect to time = d(r)/d(t)
<math>\mathrm{ =\ \langle d(x)/d(t),d(y)/d(t),d(z)/d(t) \rangle \ =\ velocity\ =\ dr/dt\ =\ dx/dt*i\ +\ dy/dt*j\ =\ x'*i\ +\ y'*j\ =\ <x',\ y'>.}</math>

Cycloid example:
<math>\vec{v}\ =\ \langle\ 1\ -\ cos(t),\ sin(t)\ \rangle </math>
at t = 0: v = 0

Speed (scalar) = |<math>\vec{v}</math>| = magnitude of v
<math>= sqrt((1-cos(t))^2+(sin(t))^2)
= sqrt((1-cos(t))^2+sin^2(t))
= sqrt(1-2*cos(t)+cos^2(t)+sin^2(t))
= sqrt(2-2*cos(t))</math>

Acceleration:
the derivative of the velocity vector. therefore the second derivative of the position vector.
<math>\vec{a} = d(\vec{v})/d(t)</math>

<math>\mathrm{e.g.\ cycloid:\ }\vec{a}\mathrm{\ =\ \langle sin(t),cos(t) \rangle }</math>
<math>\mathrm{At\ t\ =\ 0:}\vec{a}\mathrm{\ = \langle 0,1 \rangle }</math>

|d(r)/d(t)| <> d(|r|)/d(t)

differentiating the length of a vector is tricky, there is no simple formula for it, it does not need to be calculated often

Tangent vector: geometric name for d(r)/d(t) (the velocity vector). As change(t) shrinks to 0 the vector change(r)/change(t) becomes tangent to
the curve of the change in position.

When computing the vectors with a rate of time involving rotations, radians of pi should be used as a unit to represent the value of the angle
change created by the rotations. This can represent the angle variable in theta, e.g. 3 revolutions per second = 6*pi radians / sec = T.