§1.1 How do we measure velocity?

Introduction

Position and average velocity

Instantaneous Velocity

Summary

A ball's height in feet is $s(t) = 64 - 16(t-1)^2$ with $t$ in seconds, for $0 \le t \le 3$. Which description fits the graph of $y = s(t)$ on this interval?

For the ball with $s(t) = 64 - 16(t-1)^2$, compute the average velocity on the interval $[0.7, 0.8]$. Round to the nearest integer and give the number of feet per second.

A computation shows that the average velocity of the ball on $[0.4, 0.8]$ equals $12.8$ ft/sec. What does this number tell you?

Average velocities of the ball $s(t) = 64 - 16(t-1)^2$ on $[0.799, 0.8]$ and $[0.8, 0.801]$ come out to $6.416$ and $6.384$ ft/sec, and zooming in on the graph at $(0.8, s(0.8))$ makes it look almost straight. What do these observations suggest?

A ball's height in feet is $s(t) = 64 - 16(t-1)^2$ with $t$ in seconds. What is the average velocity of the ball on the interval $[1.5, 2]$, and how does it compare with the average velocity on $[0, 0.5]$?

For the ball with $s(t) = 64 - 16(t-1)^2$, estimates give an instantaneous velocity of about $-16$ ft/sec at $t = 1.5$ and about $-32$ ft/sec at $t = 2$. Which velocity is greater, and why?

If the instantaneous velocity of a moving object is negative at some time, then the object's position is decreasing at that instant.

For the same ball with $s(t) = 64 - 16(t-1)^2$, the ball reaches its highest point at $t = 1$. What is the instantaneous velocity of the ball at $t = 1$?

A falling ball has position $s(t) = 16 - 16t^2$ feet at time $t$ seconds. Which expression gives its average velocity on an interval of the form $[0.5, 0.5+h]$ with $h \ne 0$?

For the falling ball with $s(t) = 16 - 16t^2$ feet, use the average velocity expression on $[0.5, 0.5+h]$ to compute the average velocity on $[0.5, 0.75]$. Give the number of feet per second (negative for downward motion).

A bungee jumper's height is $s(t) = 100\cos(0.75t)\,e^{-0.2t} + 100$ feet. Computed average velocities include $-47.63$ ft/sec on $[0,2]$ and $-7.35$ ft/sec on $[8,10]$, while the instantaneous velocity at $t = 5$ is about $+21.78$ ft/sec. What do these signs mean?

A diver leaps from a springboard, peaks at $t = 1.1$ seconds, enters the water at $t = 2.45$, and comes to rest momentarily at the bottom of the pool before pushing off. At both the peak of the dive and the moment of resting at the bottom, the instantaneous velocity is zero.

The population of Grand Rapids was $181{,}843$ in 1980 and $197{,}800$ in 2000, with $t$ measured in years from January 1, 1980. Compute the average rate of change of the population $P$ on $[0, 20]$. Give the number of people per year, rounded to the nearest integer.