From Average Motion to Instantaneous Motion
A position-time graph shows how an object's position changes as time passes. Between two points on the graph, the slope of the line joining them describes the object's average velocity over that time interval. This joining line is called a secant line.
To describe motion at one particular instant, we shrink the time interval toward that instant. The secant slopes may approach a single value. When they do, that value is the instantaneous velocity, and it is also the slope of the tangent line to the graph at that time.
This connection gives the tangent slope a direct physical meaning. A positive or negative slope indicates the sign of the velocity; a steeper tangent indicates a larger velocity magnitude. The units come from dividing position units by time units.
Secant Slopes Are Average Velocities
Let \(s(t)\) denote an object's position at time \(t\). For two distinct times \(t_0\) and \(t_1\), its average velocity on the interval is the change in position divided by the change in time:
On the graph of position against time, the two points are \((t_0,s(t_0))\) and \((t_1,s(t_1))\). The displayed quotient is exactly the slope of the secant line through them: vertical change divided by horizontal change.
To focus on the motion near \(t_0\), write the second time as \(t_0+h\), where \(h\ne 0\) is the time increment. The corresponding average velocity, and secant slope, is
Taking \(h\) closer to zero means comparing positions at times closer to \(t_0\). The instant itself is not a time interval: \(h\) must remain nonzero while the quotient is formed.
The definition requires the displayed limit to exist as a finite number. If nearby secant slopes do not approach one finite value, this definition does not give a finite instantaneous velocity at that time. At an endpoint of the time domain, a one-sided limit may describe motion from the available side; the two-sided definition applies when times on both sides are in the domain.
Why the Limit Is Both a Velocity and a Tangent Slope
The relationship follows from two meanings of the same quotient. Its role as average velocity comes from the definition of average velocity; its role as secant slope comes from the coordinate formula for the slope of a line.
For \(h\ne 0\), compare the positions \(s(t_0)\) and \(s(t_0+h)\). Their average velocity is \(\frac{s(t_0+h)-s(t_0)}{h}\).
The two positions give graph points \((t_0,s(t_0))\) and \((t_0+h,s(t_0+h))\). The quotient is the slope of the secant through those points.
As \(h\to 0\), the second graph point approaches the first. If the secant slopes approach a finite limit, that limit defines both the instantaneous velocity and the tangent slope.
This argument is complete once the limit exists: for every nonzero \(h\), the average velocity equals the corresponding secant slope, so their limits are equal as well. The geometric description of the tangent slope as a limit of secant slopes is the key idea. It does not say that every graph has a tangent of finite slope at every point; the existence of the limit matters.
Worked Example: Reading Velocity from a Tangent Line
Worked Example: Reading Velocity from a Tangent Line
Suppose time is measured in seconds and position in metres. At \(t=4\), a position-time graph has a tangent line passing through \((2,10)\) and \((6,22)\), where each point lists time first and position second. Find the instantaneous velocity at \(t=4\).
The tangent slope is position change divided by time change:
The tangent line therefore has slope \(3\) metres per second. Since the tangent slope equals instantaneous velocity when the defining limit exists, the velocity at \(t=4\) is \(3\text{ m/s}\).
Check that the stated tangent line passes through the graph point at the indicated time. Using slope \(3\) and the point \((2,10)\), its position at \(t=4\) is \(10+3(4-2)=16\) metres. Thus the point of tangency is \((4,16)\), consistent with this line.
The positive sign means position is increasing at that instant. The line rises \(12\) metres over \(4\) seconds, which is \(3\) metres for each second of time change.
Interpreting Sign, Steepness, and Units
The sign of the tangent slope gives the sign of the instantaneous velocity. If the graph rises as time increases, its slope is positive and the velocity is positive. If it falls, the slope and velocity are negative. A horizontal tangent has slope zero, so the instantaneous velocity is zero, provided the tangent slope is defined.
The magnitude of the slope describes how quickly position is changing. A tangent with greater absolute slope corresponds to a greater velocity magnitude. For example, a slope of \(-5\) is steeper in magnitude than a slope of \(2\), because \(5>2\). The negative sign records the direction of change; it does not make the tangent less steep.
Units follow directly from the slope calculation. If position is measured in metres and time in seconds, then
If position is measured in kilometres and time in hours, the velocity is in kilometres per hour. A slope has no meaningful physical interpretation without knowing which quantity is on each graph axis and which units those quantities use.
Worked Example: Estimating a Limit from Nearby Secants
Worked Example: Estimating a Limit from Nearby Secants
A position-time graph is read near \(t=3\) seconds. Its position at that time is \(20\) metres. The nearby readings below are used to estimate the secant slopes approaching the point:
| Nearby time \(t\) (s) | Position \(s(t)\) (m) | Secant slope from \(t=3\) (m/s) |
|---|---|---|
| 2.90 | 19.61 | \(\frac{19.61-20}{2.90-3}=\frac{-0.39}{-0.10}=3.9\) |
| 2.99 | 19.9601 | \(\frac{19.9601-20}{2.99-3}=\frac{-0.0399}{-0.01}=3.99\) |
| 3.01 | 20.0401 | \(\frac{20.0401-20}{3.01-3}=\frac{0.0401}{0.01}=4.01\) |
| 3.10 | 20.41 | \(\frac{20.41-20}{3.10-3}=\frac{0.41}{0.10}=4.1\) |
The intervals are on both sides of \(t=3\). The computed secant slopes are \(3.9\), \(3.99\), \(4.01\), and \(4.1\) metres per second. The closer readings give slopes closer to \(4\) from below and above.
These finite calculations do not prove that every sufficiently nearby secant slope approaches \(4\); finitely many values alone cannot establish a limit. They support the graph-based estimate that the tangent slope, and hence the instantaneous velocity at \(t=3\), is about \(4\text{ m/s}\). The estimate is positive, indicating increasing position.
Worked Example: A Negative Tangent Slope
Worked Example: A Negative Tangent Slope
Suppose a position-time graph has a tangent at \(t=7\) seconds passing through \((5,30)\) and \((9,18)\), with time in seconds and position in metres. Find and interpret the instantaneous velocity.
The change in position is \(18-30=-12\) metres. The change in time is \(9-5=4\) seconds. Therefore the tangent slope is
Because the tangent slope at \(t=7\) is \(-3\), the instantaneous velocity there is \(-3\text{ m/s}\). Its negative sign means that position is decreasing as time increases. Its magnitude is \(3\text{ m/s}\). As a check, moving \(2\) seconds along this tangent from \(t=5\) predicts a position change of \((-3)(2)=-6\) metres, giving \(30-6=24\) metres at \(t=7\). Moving another \(2\) seconds predicts \(24-6=18\) metres at \(t=9\), agreeing with both given points.
Common Pitfall: Confusing Velocity with Speed
A negative instantaneous velocity is not a negative speed. Velocity is signed: it records whether position is increasing or decreasing. Speed is nonnegative and, at an instant, is the magnitude of velocity.
For the tangent in the previous example, writing “the speed is \(-3\text{ m/s}\)” is incorrect. The correction is: the instantaneous velocity is \(-3\text{ m/s}\), and the speed is \(|-3|=3\text{ m/s}\). A second common mistake is to treat one secant slope as the instantaneous velocity. A secant slope gives the average velocity over its interval; it gives the instantaneous velocity only in the limiting process, or when it is being used as an estimate.
Check Your Understanding
Use the connection between secant slopes, tangent slopes, and velocity to answer these questions.
- What physical quantity does the slope of a secant on a position-time graph give, and over what time period?
- State the limit that defines instantaneous velocity at a time \(t_0\), and explain what condition on the limit is needed.
- A tangent slope is \(-2.5\text{ m/s}\). What does the sign say about position, and what is the speed?
- A position-time tangent rises \(15\) metres while time increases by \(5\) seconds. Find its slope and state its units.
- Why do several nearby secant slopes support an estimate of instantaneous velocity but not, on their own, prove the limit exists?
