Skip to main content

Calculus 3 (Math 221) Notes and Study Guide

Brenton LeMesurier

Section 3.4 Motion in Space

References.

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.

Velocity, Speed and Acceleration.

Definition 3.4.1.

Velocity is the rate of change of position with respect to time as already seen with scalar valued functions, but now velocity is a vector function,
\begin{equation*} \vec{v}(t) = \vec{r}'(t). \end{equation*}
So the velocity is a tangent vector in the direction of motion.

Definition 3.4.2.

Speed is the magnitude of velocity or rate of change of position with respect to time:
\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 is as always the rate of change of velocity with respect to time, again a vector quantity
\begin{equation*} \vec{a}(t) = \vec{v}'(t) = \vec{r}''(t). \end{equation*}

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*}

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 is 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{3.4.1} \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.

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}. \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\| . \end{equation*}

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) \|} \end{equation*}

Study Guide.

Study Section 3.4 of Calculus Volume 3
 4 
openstax.org/books/calculus-volume-3/pages/3-4-motion-in-space
; in particular
  • 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.
PrevTopNext