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Section11.3The Calculus of Motion

A common use of vector–valued functions is to describe the motion of an object in the plane or in space. A position function \(\vec r(t)\) gives the position of an object at time \(t\text{.}\) This section explores how derivatives and integrals are used to study the motion described by such a function.

Definition11.3.1Velocity, Speed and Acceleration

Let \(\vec r(t)\) be a position function in \(\mathbb{R}^2\) or \(\mathbb{R}^3\text{.}\)

  1. Velocity, denoted \(\vec v(t)\text{,}\) is the instantaneous rate of position change; that is, \(\vec v(t) = \vrp(t)\text{.}\)

  2. Speed is the magnitude of velocity, \(\norm{\vec v(t)}\text{.}\)

  3. Acceleration, denoted \(\vec a(t)\text{,}\) is the instantaneous rate of velocity change; that is, \(\vec a(t) = \vec v\,'(t) = \vrp'(t)\text{.}\)

Example11.3.2Finding velocity and acceleration

An object is moving with position function \(\vec r(t) = \la t^2-t,t^2+t\ra\text{,}\) \(-3\leq t\leq 3\text{,}\) where distances are measured in feet and time is measured in seconds.

  1. Find \(\vvt\) and \vat.

  2. Sketch \vrt; plot \(\vec v(-1)\text{,}\) \(\vec a(-1)\text{,}\) \(\vec v(1)\) and \(\vec a(1)\text{,}\) each with their initial point at their corresponding point on the graph of \(\vrt\text{.}\)

  3. When is the object's speed minimized?

Example11.3.6Analyzing Motion

Two objects follow an identical path at different rates on \([-1,1]\text{.}\) The position function for Object 1 is \(\vec r_1(t) = \la t, t^2\ra\text{;}\) the position function for Object 2 is \(\vec r_2(t) = \la t^3, t^6\ra\text{,}\) where distances are measured in feet and time is measured in seconds. Compare the velocity, speed and acceleration of the two objects on the path.

Example11.3.11Analyzing the motion of a whirling ball on a string

A young boy whirls a ball, attached to a string, above his head in a counter-clockwise circle. The ball follows a circular path and makes 2 revolutions per second. The string has length 2ft.

  1. Find the position function \(\vec r(t)\) that describes this situation.

  2. Find the acceleration of the ball and derive a physical interpretation of it.

  3. A tree stands 10ft in front of the boy. At what \(t\)-values should the boy release the string so that the ball hits the tree?

Example11.3.13Analyzing motion in space

An object moves in a spiral with position function \(\vrt = \la \cos(t) , \sin(t) , t\ra\text{,}\) where distances are measured in meters and time is in minutes. Describe the object's speed and acceleration at time \(t\text{.}\)


The objects in Examples 11.3.11 and 11.3.13 traveled at a constant speed. That is, \(\norm{\vvt} = c\) for some constant \(c\text{.}\) Recall Theorem 11.2.26, which states that if a vector–valued function \(\vrt\) has constant length, then \(\vrt\) is perpendicular to its derivative: \(\vrt\cdot\vrp(t) = 0\text{.}\) In these examples, the velocity function has constant length, therefore we can conclude that the velocity is perpendicular to the acceleration: \(\vvt\cdot\vat = 0\text{.}\) A quick check verifies this.

There is an intuitive understanding of this. If acceleration is parallel to velocity, then it is only affecting the object's speed; it does not change the direction of travel. (For example, consider a dropped stone. Acceleration and velocity are parallel — straight down — and the direction of velocity never changes, though speed does increase.) If acceleration is not perpendicular to velocity, then there is some acceleration in the direction of travel, influencing the speed. If speed is constant, then acceleration must be orthogonal to velocity, as it then only affects direction, and not speed.

Key Idea11.3.14Objects With Constant Speed

If an object moves with constant speed, then its velocity and acceleration vectors are orthogonal. That is, \(\vvt\cdot\vat=0\text{.}\)

Subsection11.3.1Projectile Motion

An important application of vector–valued position functions is projectile motion: the motion of objects under only the influence of gravity. We will measure time in seconds, and distances will either be in meters or feet. We will show that we can completely describe the path of such an object knowing its initial position and initial velocity (i.e., where it is and where it is going.)

Suppose an object has initial position \(\vec r(0) = \la x_0,y_0\ra\) and initial velocity \(\vec v(0) = \la v_x,v_y\ra\text{.}\) It is customary to rewrite \(\vec v(0)\) in terms of its speed \(v_0\) and direction \(\vec u\text{,}\) where \(\vec u\) is a unit vector. Recall all unit vectors in \(\mathbb{R}^2\) can be written as \(\la \cos(\theta) ,\sin(\theta) \ra\text{,}\) where \(\theta\) is an angle measure counter–clockwise from the \(x\)-axis. (We refer to \(\theta\) as the angle of elevation.) Thus \(\vec v(0) = v_0\la \cos(\theta) ,\sin(\theta) \ra\text{.}\)

Since the acceleration of the object is known, namely \(\vat = \la 0,-g\ra\text{,}\) where \(g\) is the gravitational constant, we can find \(\vrt\) knowing our two initial conditions. We first find \(\vvt\text{:}\)

In this text we use \(g=32\)ft/s when using Imperial units, and \(g=9.8\)m/s when using SI units.

\begin{align*} \vec v(t) \amp = \int \vat \ dt\\ \vvt \amp = \int \la 0,-g\ra \ dt\\ \vvt \amp = \la 0,-gt\ra + \vec C. \end{align*}

Knowing \(\vec v(0) = v_0\la \cos(\theta) ,\sin(\theta) \ra\text{,}\) we have \(\vec C = v_0\la \cos(t) ,\sin(t) \ra\) and so \begin{equation*} \vec v(t) = \la v_0\cos(\theta) , -gt+v_0\sin(\theta) \ra. \end{equation*}

We integrate once more to find \(\vrt\text{:}\) \begin{align*} \vrt \amp = \int \vvt\ dt\\ \vrt \amp = \int \la v_0\cos(\theta) , -gt+v_0\sin(\theta) \ra\ dt\\ \vrt \amp = \la \big(v_0\cos(\theta) \big)t, -\frac12gt^2+\big(v_0\sin(\theta) \big)t\ra + \vec C.\\ \end{align*} Knowing \(\vec r(0) = \la x_0,y_0\ra\text{,}\) we conclude \(\vec C = \la x_0,y_0\ra\) and \begin{align*} \vrt \amp = \la \big(v_0\cos(\theta) \big)t+x_0\ , -\frac12gt^2+\big(v_0\sin(\theta) \big)t+y_0\ \ra. \end{align*}

Key Idea11.3.15Projectile Motion

The position function of a projectile propelled from an initial position of \(\vec r_0=\la x_0,y_0\ra\text{,}\) with initial speed \(v_0\text{,}\) with angle of elevation \(\theta\) and neglecting all accelerations but gravity is \begin{equation*} \vrt = \la \big(v_0\cos(\theta) \big)t+x_0\ , -\frac12gt^2+\big(v_0\sin(\theta) \big)t+y_0\ \ra. \end{equation*}

Letting \(\vec v_0 = v_0\la \cos(\theta) ,\sin(\theta) \ra\text{,}\) \(\vrt\) can be written as \begin{equation*} \vrt = \la 0,-\frac12gt^2\ra + \vec v_0t+\vec r_0. \end{equation*}

We demonstrate how to use this position function in the next two examples.

Example11.3.16Projectile Motion

Sydney shoots her Red Ryder\textregistered bb gun across level ground from an elevation of 4ft, where the barrel of the gun makes a \(5^\circ\) angle with the horizontal. Find how far the bb travels before landing, assuming the bb is fired at the advertised rate of 350ft/s and ignoring air resistance.

Example11.3.17Projectile Motion

Alex holds his sister's bb gun at a height of 3ft and wants to shoot a target that is 6ft above the ground, 25ft away. At what angle should he hold the gun to hit his target? (We still assume the muzzle velocity is 350ft/s.)


Subsection11.3.2Distance Traveled

Consider a driver who sets her cruise–control to 60mph, and travels at this speed for an hour. We can ask:

  1. How far did the driver travel?

  2. How far from her starting position is the driver?

The first is easy to answer: she traveled 60 miles. The second is impossible to answer with the given information. We do not know if she traveled in a straight line, on an oval racetrack, or along a slowly–winding highway.

This highlights an important fact: to compute distance traveled, we need only to know the speed, given by \(\norm{\vvt}\text{.}\)

Note that this is just a restatement of Theorem 11.2.30: arc length is the same as distance traveled, just viewed in a different context.

Example11.3.19Distance Traveled, Displacement, and Average Speed

A particle moves in space with position function \(\vrt = \la t,t^2,\sin(\pi t)\ra\) on \([-2,2]\text{,}\) where \(t\) is measured in seconds and distances are in meters. Find:

  1. The distance traveled by the particle on \([-2,2]\text{.}\)

  2. The displacement of the particle on \([-2,2]\text{.}\)

  3. The particle's average speed.


In Definition 5.4.26 of Chapter 5 we defined the average value of a function \(f(x)\) on \([a,b]\) to be \begin{equation*} \frac{1}{b-a}\int_a^bf(x)\ dx. \end{equation*}

Note how in Example 11.3.19 we computed the average speed as \begin{equation*} \frac{\text{ distance traveled } }{\text{ travel time } } = \frac1{2-(-2)}\int_{-2}^2\norm{\vvt}\ dt; \end{equation*} that is, we just found the average value of \(\norm{\vvt}\) on \([-2,2]\text{.}\)

Likewise, given position function \(\vrt\text{,}\) the average velocity on \([a,b]\) is \begin{equation*} \frac{\text{ displacement } }{\text{ travel time } } = \frac1{b-a}\int_a^b \vec{r}\,'(t)\ dt = \frac{\vec r(b)-\vec r(a)}{b-a}; \end{equation*} that is, it is the average value of \(\vec r\,'(t)\text{,}\) or \(\vvt\text{,}\) on \([a,b]\text{.}\)

Key Idea11.3.21Average Speed, Average Velocity

Let \(\vec r(t)\) be a continuous position function on an open interval \(I\) containing \(a\lt b\text{.}\)

The average speed is: \begin{equation*} \frac{\text{ distance traveled } }{\text{ travel time } } = \frac{\int_a^b \norm{\vvt}\ dt}{b-a} = \frac1{b-a}\int_a^b\norm{\vvt}\ dt. \end{equation*}

The average velocity is: \begin{equation*} \frac{\text{ displacement } }{\text{ travel time } } = \frac{\int_a^b \vec{r}\,'(t)\ dt}{b-a} = \frac1{b-a}\int_a^b\vec{r}\,'(t)\ dt. \end{equation*}

The next two sections investigate more properties of the graphs of vector–valued functions and we'll apply these new ideas to what we just learned about motion.


In the following exercises, a position function \(\vrt\) is given. Find \(\vvt\) and \(\vat\text{.}\)

In the following exercises, a position function \(\vrt\) is given. Sketch \(\vrt\) on the indicated interval. Find \(\vvt\) and \(\vat\text{,}\) then add \(\vec v(t_0)\) and \(\vec a(t_0)\) to your sketch, with their initial points at \(\vec r(t_0)\text{,}\) for the given value of \(t_0\text{.}\)

In the following exercises, a position function \(\vrt\) of an object is given. Find the speed of the object in terms of \(t\text{,}\) and find where the speed is minimized/maximized on the indicated interval.

In the following exercises, position functions \(\vec r_1(t)\) and \(\vec r_2(s)\) for two objects are given that follow the same path on the respective intervals.

  1. Show that the positions are the same at the indicated \(t_0\) and \(s_0\) values; i.e., show \(\vec r_1(t_0) = \vec r_2(s_0)\text{.}\)

  2. Find the velocity, speed and acceleration of the two objects at \(t_0\) and \(s_0\text{,}\) respectively.

In the following exercises, find the position function of an object given its acceleration and initial velocity and position.

In the following exercises, find the displacement, distance traveled, average velocity and average speed of the described object on the given interval.

The following exercises ask you to solve a variety of problems based on the principles of projectile motion.