Definition 3.4.1.
Velocity
\begin{equation*}
\vec{v}(t) = \vec{r}'(t).
\end{equation*}
Section 3.4 Motion in Space
References.
- OpenStax Calculus Volume 3, Section 3.4.
1
openstax.org/books/calculus-volume-3/pages/3-4-motion-in-space - Calculus, Early Transcendentals by Stewart, Section 13.4.
Introduction.
Many of the geometrical ideas from the previous section relate to physical quantities like speed, velocity and acceleration, in the natural case where the value of a function \(\vec{r}(t)\) is a position space and the value of its argument \(t\) is time.
Topics.
Subsection 3.4.1 Velocity, Speed and Acceleration
Definition 3.4.2.
Speed
\begin{equation*}
v = \| \vec{v}(t) \| = \| \vec{r}'(t) \| = \frac{ds}{dt}.
\end{equation*}
Note that arc length \(s\) has the natural physical meaning of distance traveled along the curve.
Definition 3.4.3.
Acceleration
\begin{equation*}
\vec{a}(t) = \vec{v}'(t) = \vec{r}.
\end{equation*}
Remark 3.4.4.
With this new notation, the formulas (3.3.6) and (3.3.12) for the curvature and pricipal unit normal can be expressed as
\begin{equation}
\kappa = \frac{\| \vec{v} \times \vec{a} \|}{v^3}\tag{3.4.1}
\end{equation}
and
\begin{equation}
\N = \frac{1}{v \kappa}\frac{d \T}{dt}
= \frac{v^2}{\|\vec{v} \times \vec{a}\|}\frac{d \T}{dt}\tag{3.4.2}
\end{equation}
Subsection 3.4.2 Force and Motion: Position from Acceleration
Newton’s laws relate motion to the force acting on a body of mass \(m\text{,}\) with the force a vector \(\vec{F}\) and determining acceleration through
\begin{equation*}
\vec{F} = m \vec{a}.
\end{equation*}
If the force is known as a function of time, along with the position \(\vec{r}_0\) and velocity \(\vec{v}_0\) at one time \(t_0\text{,}\) the position at all times can be calculated:
\begin{equation*}
\vec{a} = \vec{F}/m, \quad \vec{v}(t) = \vec{v}_0 + \int_{t_0}^t \vec{a}(u) du, \quad \vec{r}(t) = \vec{r}_0 + \int_{t_0}^t \vec{v}(u) du.
\end{equation*}
Subsection 3.4.3 Tangential and Normal Components of Acceleration
With motion along a straight line, acceleration can be thought of as rate of change of speed. With motion in a plane or space, acceleration can also change direction, perhaps with no change in speed.
These two effects of acceleration can be described in terms of the tangential and normal components of acceleration.
First, compute the speed \(v=\|\vecv\|\text{,}\) so that \(\T = \vecv/v\) and
\begin{equation*}
\vec{v} = v\T.
\end{equation*}
Then
\begin{equation*}
\vec{a}= \vecv' = \big( v \T \big)' = v'\T + v\T'\text{.}
\end{equation*}
Also \(\kappa = \|\T'\|/\|\vec{r}'\| = \|\T'\|/v,\) so
\begin{equation*}
\| \T' \| = \kappa v
\end{equation*}
and \(\N=\T'/\|\T'\|\) so
\begin{equation*}
\T' = \|\T'\|\N = \kappa v \N.
\end{equation*}
All together,
\begin{equation}
\vec{a} = v' \T + \kappa v^2 \N, = a_T \T + a_N \N.\tag{3.4.3}
\end{equation}
Here \(a_T=v'\) and \(a_N = \kappa v^2\) are the tangential and normal components of the acceleration:
- the tangential component \(a_T\) measures change of speed;
- the normal component \(a_N\) measures change of direction.
Note that there is no "binormal component of acceleration": instantaneously, acceleration acts in the osculating plane. From another perspective, the osculating plane at any time is the plane determined by the current position, velocity and acceleration, in which the motion appears momentarily to lie.
Subsection 3.4.4 Acceleration Components in Terms of Velocity and Acceleration Vectors
It is often convenient to express everything in terms of the basic derivatives of components of \(\vec{r}(t)\text{,}\) without use of intermediate quantities like \(v'(t)\) and \(\kappa(t)\) thus avoiding differentiation of square roots. First it can be checked that \(\vec{v} \cdot \vec{a} = v v'\text{,}\) so that
\begin{equation}
a_T = v' = \frac{\vec{v} \cdot \vec{a}}{v} = \frac{\vec{r}' \cdot \vec{r}''}{\|\vec{r}'\|},
\quad = \frac{d\vec{r}}{ds} \cdot \frac{d^2\vec{r}}{ds^2}.\tag{3.4.4}
\end{equation}
Next, recalling from Theorem 3.3.3 that \(\kappa = {\| \vec{r}'(t) \times \vec{r}''(t) \|}/{\| \vec{r}'(t) \|^3}\text{,}\)
\begin{equation}
a_N = \kappa v^2 = \frac{\|\vec{r}'(t) \times \vec{r}''(t)\|}{\|\vec{r}'(t)\|},
\quad = \left\| \frac{d\vec{r}}{ds} \times \frac{d^2\vec{r}}{ds^2} \right\| .\tag{3.4.5}
\end{equation}
Subsection 3.4.5 Another Way to Compute \(\N\)
These formulas for \(a_T\) and \(a_N\) also give a sometimes more convenient way to compute the normal vector:
\begin{equation}
\N = \frac{\vec{a} - a_T \T}{a_N} = \frac{\vec{r}''-(a_T/v) \vec{r}'}{a_N}
= \frac{v \vec{r}''- v' \vec{r}'}{\| \vec{r}'(t) \times \vec{r}''(t) \|}
= \frac{v \vec{a} - v' \vec{v}}{\| \vec{v} \times \vec{a} \|}\tag{3.4.6}
\end{equation}
Study Guide.
Study Section 3.4 of Calculus Volume 3; in particular
4
openstax.org/books/calculus-volume-3/pages/3-4-motion-in-space- The Definitions.
- Theorems 3.7 and 3.8.
- Examples 3.14 and 3.15, and the Checkpoints following each.
- Several exercises from the range 157–162.
Note: The topics after Checkpoint 3.15, Projectile Motion and Kepler’s Laws should be interesting reading for some students, but will not be covered on quizzes or tests.
