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Section Average Speed and Velocity

The average speed of an object that has traveled a distance d in time t is v = d/t. The terms velocity and speed are interchangeable except that velocity may be negative to indicate direction. For example, if a rock is thrown in the air, we would typically assign a positive velocity on the way up and a negative velocity on the way down.

Problem 1.1.

A car goes from Houston, Texas to Saint Louis, Missouri (a distance of 600 miles). The trip takes 9 hours and 28 minutes. What was the average speed of the trip? What is one true statement that you can make about the time(s) (if any) at which the car moved at this speed?

Problem 1.2.

Robbie the robot is walking down a road in a straight line, away from a pothole. Table 1.3 gives the distance between Robbie and the pothole at different times, where the time is measured in seconds and distance is measured in feet. Write a formula that expresses Robbie's distance from the pothole as a function of time. Use \(R\) to represent distance and \(t\) to represent time in your formula. This function will be called Robbie's position function.

Time (sec) Position (feet)
\(0\) \(2\)
\(1\) \(3\)
\(2\) \(10\)
\(3\) \(29\)
\(4\) \(66\)
\(5\) \(127\)
Table 1.3. Position Table
Definition 1.4.

A relation is a collection of points in the plane.

Definition 1.5.

A function is a collection of points in the plane with the property that no two points have the same first coordinates. The set of all first coordinates of these points is referred to as the domain of the function and the set of all second coordinates of these points is referred to as the range of the function.

Example 1.6.

Functions may be expressed in many different forms.

The function

\begin{equation*} f = \{ (x,y) : y = 6x + 2 \} = \{ \dots , (-1,-4), (0, 2), (1,8), (\pi, 6\pi+2), \dots \} \end{equation*}

might be expressed via the short-hand,

\begin{equation*} y = 6x + 2 \; \; \mbox{ or } \; \; f(x) = 6x +2 \; \; \mbox{ or } \; \; f(z) = 6z + 2. \end{equation*}

The function is not the written equation, rather it is the set of points in the plane that satisfy the equation. While every function is a relation, every relation is not necessarily a function. Examples of relations that are not functions would be the vertical line,

\begin{equation*} \{ (x,y) : x = 6 \}\; \; \mbox{written in short-hand as} \; \; x=6, \end{equation*}

and the circle,

\begin{equation*} \{ (x,y) : x^2 + y^2 = 9 \} \; \; \mbox{written in short-hand as} \; \;x^2 + y^2 =9. \end{equation*}
Problem 1.7.

Express Robbie's position function in the three different forms just discussed. Graph Robbie's position function, labeling the axes and several points.

Interval and Set Notation. We use \(\re\) to denote the set of all real numbers and \(x \in S\) to mean that \(x\) is a member of the set \(S.\) If \(a \in \re\) and \(b \in \re\) and \(a \lt b\) then the open interval \((a,b)\) denotes the set of all numbers between \(a\) and \(b.\) For example,

  1. \((a,b) = \{ x \in \re : a \lt x \lt b \}\) — an open interval

  2. \([a,b] = \{ x \in \re : a \leq x \leq b \}\) — a closed interval

  3. \((a,b] = \{ x \in \re : a \lt x \leq b \}\) — a half-open interval

When convenient, we'll use the notation \((-\infty, b)\) to mean all numbers less than \(b,\) even though \(-\infty\) is not a real number. We won't define \(\infty\) or \(-\infty\) in this class, although you can read a bit about them in the Extended Reals .

Problem 1.8.

Sketch the graph of each of the following and state the domain and range of each.

  1. \(\dsp f(x) = \frac{1}{x-1}\)

  2. \(g(x) = \sqrt{4-x}\)

  3. \(x = y^2 + 1\)

  4. \(i(x) = -(x-2)^3 +4\)

The next several problems deal with Robbie's motion where \(R(t)\) represents Robbie's distance from the pothole in feet at time \(t\) in seconds.

Problem 1.9.

Assume c is a positive number and compute \(R(3)\) and \(R(c).\) Write a sentence that says what \(R(3)\) means. What does \(R(c)\) mean?

Problem 1.10.

Compute the value of \(R(5) - R(3).\) What is the physical (real world, useful) meaning of this number? Assume u and v are positive numbers and compute \(R(u) - R(v).\) What is the physical meaning of \(R(u)-R(v)?\)

Problem 1.11.

Evaluate each of the following ratios and explain in detail their physical meaning. Assume that \(u\) and \(v\) are positive numbers and \(u > v.\)

  1. \(\displaystyle{ \frac{R(5)-R(3)}{5-3}}\)

  2. \(\displaystyle{ \frac{R(u)-R(v)}{u-v}}\)

Problem 1.12.

Assuming that the function you found is valid for all times \(t > 4,\) where is Robbie when the time is \(t=5\) seconds? \(t=6\) seconds? \(t=5.5\) seconds? When is Robbie \(106\) feet from the pothole?

Problem 1.13.

Recall from Problem 1.1 that the distance from Houston, Texas to Saint Louis, Missouri is 600 miles. A car goes from Houston to Saint Louis traveling at 65 mph for the first 22 minutes, accelerates to 95 mph, and remains at that speed for the rest of the trip. Estimate the average speed of the trip. Explain whether your answer is exact or an approximation.

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