14.2 Motion
In order to study motion, we'll need to take time derivatives of vectors.
Laws.
The derivative law for addition states
\[
\begin{align}
\overset{t}{\diff} \bigl(v + w\bigr) &= \overset{t}{\diff} v + \overset{t}{\diff} w
\end{align}
\]
and the derivative law for multiplication states
\[
\begin{align}
\overset{t}{\diff} \bigl(n \mult v\bigr) &= \overset{t}{\diff} n \mult v + n \mult \overset{t}{\diff} v
\end{align}
\]
where
\[
\begin{align}
v &: \Vector_{},
&
w &: \Vector_{}
\end{align}
\]
are vectors and \( n : \Number \) is a number.
The standard vectors have zero derivative.
\[
\begin{align}
\overset{t}{\diff} \bigl(\vec{\standard}_{x}\bigr) &= 0
&
&\andSpaced
&
\overset{t}{\diff} \bigl(\vec{\standard}_{y}\bigr) &= 0
\end{align}
\]
The standard vectors therefore pull out of the derivative
\[
\begin{align}
\overset{t}{\diff} \bigl(n \mult \vec{\standard}_{x}\bigr) &= \overset{t}{\diff} n \mult \vec{\standard}_{x}
\\
\overset{t}{\diff} \bigl(n \mult \vec{\standard}_{x}\bigr) &= \overset{t}{\diff} n \mult \vec{\standard}_{y}
\end{align}
\]
where \( n : \Number \) is any number.
Position, Velocity, and Speed
A path
\[
\begin{align}
p &: \Path_{(x, y)}
\\
&\left\{\begin{aligned} x_p &\depends t \\ y_p &\depends t \end{aligned}\right.
\end{align}
\]
has position vector, velocity vector, and speed as follows.
\[
\begin{align}
\position p &: \Vector_{(x, y)}
&
\velocity p &: \Vector_{(x, y)}
&
\speed p &: \Number
\\
\position p &= x_p \mult \vec{\standard}_{x} + y_p \mult \vec{\standard}_{y}
&
\velocity p &= \overset{t}{\diff} \position p
&
\speed p &= \magnitude \velocity p
\end{align}
\]
The velocity vector points in the direction of travel and so is sometimes called the tangent vector. But the velocity tells us more than just a direction: its magnitude tells us how fast our tracing point moves!
Example.
Let's revisit a path that we looked at in the previous section. Consider
\[
\begin{align}
p &: \Path_{(x, y)}
\\
&\left\{\begin{aligned} x_p &= t^3 - t \\ y_p &= t^2 - 1 \end{aligned}\right.
\end{align}
\]
for the time interval
\[
\begin{align}
i &: \Interval_{t}
\\
i &= [-2, 2]_{t}.
\end{align}
\]
This path has position, velocity, and speed as follows.
\[
\begin{align}
\position p &= (t^3 - t) \mult \vec{\standard}_{x} + (t^2 - 1) \mult \vec{\standard}_{y}
\\
\velocity p &= (3 t^2 - 1) \mult \vec{\standard}_{x} + 2 t \mult \vec{\standard}_{y}
\\
\speed p &= \sqrt[2]{9 t^4 - 2 t^2 + 1}
\end{align}
\]
Let's look at these for a few \( t \)-values.
| \( t \) | \( \position p \) | \( \velocity p \) | \( \speed p \) |
|---|---|---|---|
| \( -2 \) | \( -6 \mult \vec{\standard}_{x} + 3 \mult \vec{\standard}_{y} \) | \( 11 \mult \vec{\standard}_{x} - 4 \mult \vec{\standard}_{y} \) | \( \sqrt[2]{137} \) |
| \( -1 \) | \( 0 \) | \( 2 \mult \vec{\standard}_{x} - 2 \mult \vec{\standard}_{y} \) | \( \sqrt[2]{8} \) |
| \( 0 \) | \( - \vec{\standard}_{y} \) | \( - \vec{\standard}_{x} \) | \( 1 \) |
| \( 1 \) | \( 0 \) | \( 2 \mult \vec{\standard}_{x} + 2 \mult \vec{\standard}_{y} \) | \( \sqrt[2]{8} \) |
| \( 2 \) | \( 6 \mult \vec{\standard}_{x} + 3 \mult \vec{\standard}_{y} \) | \( 11 \mult \vec{\standard}_{x} + 4 \mult \vec{\standard}_{y} \) | \( \sqrt[2]{137} \) |
The velocity vector at time \( t = 0 \) points due West. The velocity at times, \( t = -1 \) and \( t = 1 \), points South-East and North-East, respectively.
Compared with the other times in our table, the tracing point is moving relatively slowly at time \( t = 0 \). That said, we can find times when the tracing point moves slower: you can check that the lowest speed
\[
\begin{align}
\speed p &= \sqrt[2]{\frac{8}{9}}
\end{align}
\]
occurs at times \( t = -1/3 \) and \( t = 1/3 \). So oddly enough, \( t = 0 \) is a high point for speed!