Notes on 1.1: How do we measure velocity?
1.1 How do we measure velocity? — Comprehensive Notes
Key motivating questions
How is the average velocity of a moving object connected to the values of its position function?
How do we interpret the average velocity geometrically on the graph of the position function?
How is instantaneous velocity connected to average velocity?
1.1.1 Introduction
Calculus as the study of change: focus on how fast quantities change.
A natural problem: a ball thrown straight up in the air – how is the ball moving?
Preview activities illustrate the link between position, average velocity, and instantaneous velocity.
Preview function example (from Preview Activity 1.1.1): height of a ball at time t (seconds) given by the formula
s(t) = 64 - 16(t - 1)^2.
Time: 0 ≤ t ≤ 3 (units: seconds), height in feet.
The discussion motivates the development of the concept of velocity from changes in position over time.
1.1.2 Position and Average Velocity
A moving object has a position that is a function of time; for straight-line motion, the position is a single variable, denoted by s(t).
Example: the position might be the mile marker of a car on a straight highway, as a function of time in hours.
On a given time interval, the average velocity is defined as the rate of change of position over that interval.
Definition (Average Velocity):
For an object moving in a straight line with position function s(t), the average velocity on the interval from t = a to t = b (with a < b) is
ext{AV}[a,b] = rac{s(b) - s(a)}{b - a}.
Units: the units on ext{AV} are "units of s per unit of t" (e.g., feet per second, miles per hour).
Geometric interpretation: ext{AV}[a,b] is the slope of the line connecting the points \( (a, s(a)) \) and \( (b, s(b)) \) on the graph of s(t) (the secant line).
Preview activity uses the given function s(t) = 64 - 16(t - 1)^2 to illustrate how to compute average velocity on chosen intervals.
Example (from Preview Activity): on the interval from a = 0.5 to b = 1,
ext{AV}[0.5,1] = rac{s(1) - s(0.5)}{1 - 0.5}.
The units are feet per second (ft/s) for height in feet and time in seconds.
1.1.3 Instantaneous Velocity
Intuition: even though a velocity value is defined at each moment, we can think of instantaneous velocity as the limit of average velocities as the time interval around a fixed time becomes arbitrarily small.
Informal definition: The instantaneous velocity at time t = a is the value that the average velocity approaches as we consider smaller and smaller time intervals containing a.
The precise development (to come) uses the limit of average velocities as the interval shrinks to zero.
Practical interpretation: a car’s speedometer gives an instantaneous velocity, which is essentially an average velocity over a very small time interval around the instant in question.
Conceptual link: from average velocity on [a, a+h] to the instantaneous velocity at t=a by letting h o 0.$n
1.1.2 (Preview) and 1.1.4 (Worked practice) highlights
For the given height function s(t) = 64 - 16(t - 1)^2, you can:
Compute the average velocity on various intervals like \( [0.5, 1], [0, 0.5], [1, 3] \).
Interpret the slope of the secant line on the graph of s(t) as the average velocity on that interval.
A fundamental activity is to examine how the line through \( (a, s(a)) \) and \( (b, s(b)) \) behaves and what its slope (the average velocity) says about the ball’s motion on that interval.
The instantaneous velocity is approached by drawing secant lines for smaller and smaller h around the time of interest and observing the resulting slopes.
Summary of core ideas and connections
Average velocity on an interval is the average rate of change of the position function over that interval:
ext{AV}[a,b] = rac{s(b) - s(a)}{b - a}.
Geometrically, this is the slope of the secant line joining the two points on the position graph.
Instantaneous velocity is the limit of average velocities as the interval shrinks to the instantaneous time:
v(a) = ext{lim}_{h o 0} rac{s(a+h) - s(a)}{h}.
The instantaneous velocity is the slope of the tangent line to the graph of the position function at time a (provided the derivative exists).
The sign of the velocity tells you the direction of motion; the magnitude tells you how fast (speed).
Real-world interpretation: speedometers measure instantaneous velocity approximately by using a tiny time window to estimate the average velocity.
Notation recap and definitions (quick reference)
Position function: s(t) (units: e.g., feet, meters; time unit: seconds, hours)
Time interval: [a,b] with a < b
Average velocity on [a,b]: ext{AV}[a,b] = rac{s(b) - s(a)}{b - a}
Slope interpretation: slope of the secant line through \( (a, s(a)) \) and \( (b, s(b)) \)
Instantaneous velocity at time a: v(a) = ext{lim}_{h o 0} rac{s(a+h) - s(a)}{h}
If the derivative exists, v(a) = s'(a) and is the slope of the tangent line to the graph of s(t) at t=a.
Connections to broader principles
The move from average to instantaneous velocity mirrors the core idea of derivatives as rates of change.
The geometric interpretation (secant vs. tangent) emphasizes the relationship between algebraic formulas and graphical intuition.
This framework generalizes beyond vertical motion to any motion along a straight line where the position is given by a function of time.
Key formulas to memorize (LaTeX)
Average velocity on \[a,b]:
ext{AV}[a,b] = rac{s(b) - s(a)}{b - a}.
Secant slope interpretation: the line joining \( (a, s(a)) \) and \( (b, s(b)) \) has slope ext{AV}[a,b].
Instantaneous velocity (limit):
v(a) = ext{lim}_{h \to 0} \frac{s(a+h) - s(a)}{h}.
If s(t) is differentiable at t=a, then v(a) = s'(a).
Quick worked checks (using the chosen example)
For s(t) = 64 - 16(t-1)^2:
s(0) = 48,\ s(0.5) = 60,\ s(1) = 64,\ s(1.5) = 60,\ s(2) = 48,\ s(3) = 0.
ext{AV}[0,3] = \frac{0 - 48}{3} = -16\text{ ft/s}.
ext{AV}[0.5,1] = \frac{64 - 60}{0.5} = 8\text{ ft/s}.
ext{AV}[0,0.5] = \frac{60 - 48}{0.5} = 24\text{ ft/s}.
If we differentiate: s'(t) = -32t + 32\implies v(1) = s'(1) = 0\text{ ft/s}.$$
Final takeaway
Velocity measures how fast the position changes with time.
Average velocity gives a coarse measure over an interval; instantaneous velocity is the refined measure at a single moment, found as a limit and represented by the derivative when it exists.