§1.1 How do we measure velocity?

Introduction

Position and average velocity

Instantaneous Velocity

Summary

Suppose that the height $s$ of a ball at time $t$ (in seconds) is given in feet by the formula $s(t) = 64 - 16(t-1)^2$.

Construct a graph of $y = s(t)$ on the time interval $0 \le t \le 3$. Label at least six distinct points on the graph, including the three points showing when the ball was released, when the ball reaches its highest point, and when the ball lands.

Compute the average velocity of the ball on each of the following time intervals: $[0.4,0.8]$, $[0.7,0.8]$, $[0.79, 0.8]$, $[0.799,0.8]$, $[0.8,1.2]$, $[0.8,0.9]$, $[0.8,0.81]$, $[0.8,0.801]$. Include units for each value.

On the graph provided in (a), sketch the line that passes through the points $A=(0.4, s(0.4))$ and $B=(0.8, s(0.8))$. What is the meaning of the slope of this line? In light of this meaning, what is a geometric way to interpret each of the values computed in the preceding question?

Use a graphing utility to plot the graph of $s(t) = 64 - 16(t-1)^2$ on an interval containing the value $t = 0.8$. Then, zoom in repeatedly on the point $(0.8, s(0.8))$. What do you observe about how the graph appears as you view it more and more closely?

What do you conjecture is the velocity of the ball at the instant $t = 0.8$? Why?

Compute the average velocity of the ball on the time interval $[1.5,2]$. What is different between this value and the average velocity on the interval $[0,0.5]$?

Use appropriate computing technology to estimate the instantaneous velocity of the ball at $t = 1.5$. Likewise, estimate the instantaneous velocity of the ball at $t = 2$. Which value is greater?

How is the sign of the instantaneous velocity of the ball related to its behavior at a given point in time? That is, what does positive instantaneous velocity tell you the ball is doing? Negative instantaneous velocity?

Without doing any computations, what do you expect to be the instantaneous velocity of the ball at $t = 1$? Why?

The position function for a falling ball is given by $s(t) = 16 - 16t^2$ (where $s$ is measured in feet and $t$ in seconds).

• Find an expression for the average velocity of the ball on a time interval of the form $[0.5, 0.5+h]$ where $-0.5 \lt h \lt 0.5$ and $h \ne 0$. - Use this expression to compute the average velocity on $[0.5,0.75]$ and $[0.4,0.5]$. - Make a conjecture about the instantaneous velocity at $t = 0.5$.

A bungee jumper dives from a tower at time $t=0$. Her height $s$ (measured in feet) at time $t$ (in seconds) is given by the graph in Figure 1.1. In this problem, you may base your answers on estimates from the graph or use the fact that the jumper's height function is given by $s(t) = 100\cos(0.75t) \cdot e^{-0.2t}+100$.

• What is the change in vertical position of the bungee jumper between $t=0$ and $t=15$? - Estimate the jumper's average velocity on each of the following time intervals: $[0,15]$, $[0,2]$, $[1,6]$, and $[8,10]$. Include units on your answers. - On what time interval(s) do you think the bungee jumper achieves her greatest average velocity? Why? - Estimate the jumper's instantaneous velocity at $t=5$. Show your work and explain your reasoning, and include units on your answer. - Among the average and instantaneous velocities you computed in earlier questions, which are positive and which are negative? What does negative velocity indicate?

A diver leaps from a 3 meter springboard. Their feet leave the board at time $t=0$, they reaches a maximum height of 4.5 m at $t = 1.1$ seconds, and enter the water at $t = 2.45$. Once in the water, the diver coasts to the bottom of the pool (depth 3.5 m), touches bottom at $t=7$, rests for one second, and then pushes off the bottom. From there they coast to the surface, and take a breath at $t=13$.

• Let $s(t)$ denote the function that gives the height of the diver's feet (in meters) above the water at time $t$. (Note that the “height” of the bottom of the pool is $-3.5$ meters.) Sketch a carefully labeled graph of $s(t)$ on the provided axes in Figure 1.1. Include scale and units on the vertical axis. Be as detailed as possible. Axes for plotting $s(t)$ in part (a). Axes for plotting $v(t)$ in part (c). - Based on your graph in (a), what is the average velocity of the diver between $t = 2.45$ and $t=7$? Is their average velocity the same on every time interval within $[2.45,7]$? - Let the function $v(t)$ represent the instantaneous vertical velocity of the diver at time $t$ (i.e. the rate at which the height function $s(t)$ is changing; note that velocity in the upward direction is positive, while the velocity of a falling object is negative). Based on your understanding of the diver's behavior, as well as your graph of the position function, sketch a carefully labeled graph of $v(t)$ on the axes provided in Figure 1.1. Include scale and units on the vertical axis. Write several sentences that explain how you constructed your graph, discussing when you expect $v(t)$ to be zero, positive, negative, relatively large, and relatively small. - Is there a connection between the two graphs that you can describe? What can you say about the velocity graph when the height function is increasing? decreasing? Make as many observations as you can.

According to the U.S. census, the population of the city of Grand Rapids, MI, was 181,843 in 1980; 189,126 in 1990; and 197,800 in 2000.

• Between 1980 and 2000, by how many people did the population of Grand Rapids grow? - In an average year between 1980 and 2000, by how many people did the population of Grand Rapids grow? - Just like we can find the average velocity of a moving body by computing change in position over change in time, we can compute the average rate of change of any function $f$. In particular, the average rate of change of a function $f$ over an interval $[a,b]$ is the quotient

. What does the quantity $\frac{f(b)-f(a)}{b-a}$ measure on the graph of $y = f(x)$ over the interval $[a,b]$? - Let $P(t)$ represent the population of Grand Rapids at time $t$, where $t$ is measured in years from January 1, 1980. What is the average rate of change of $P$ on the interval $t = 0$ to $t = 20$? What are the units on this quantity? - If we assume the population of Grand Rapids is growing at a rate of approximately 4% per decade, we can model the population function with the formula

. Use this formula to compute the average rate of change of the population on the intervals $[5,10]$, $[5,9]$, $[5,8]$, $[5,7]$, and $[5,6]$. - How fast do you think the population of Grand Rapids was changing on January 1, 1985? Said differently, at what rate do you think people were being added to the population of Grand Rapids as of January 1, 1985? How many additional people should the city have expected in the following year? Why?