Instantaneous Velocity and Tangent Slopes

From Average Velocity to Instantaneous Velocity

Average velocity describes the change in position over a time interval. If an object has position \(s(t)\) at time \(t\), its average velocity from \(t=a\) to \(t=b\), where \(a\ne b\), is the change in position divided by the elapsed time:

This gives one rate for the entire interval. It does not generally tell us the object's velocity at a particular instant within that interval. To describe motion at exactly time \(t\), we examine average velocities over intervals that begin at \(t\) and become shorter.

The quotient inside this limit is the average velocity from time \(t\) to time \(t+h\). Thus, instantaneous velocity is not an average over a nonzero time interval; it is the limit of those average velocities as the interval shrinks toward the instant.

If position is measured in meters and time in seconds, then \(s'(t)\) has units of meters per second. More generally, derivative units are position units divided by time units. The value is a signed rate: its sign and size belong to the velocity, not necessarily to the object's speed.

Why Velocity Is a Tangent Slope

On a position-time graph, time is the horizontal coordinate and position is the vertical coordinate. Pick two points on the curve, at times \(t\) and \(t+h\). The line through them is a secant line. Its slope is

That slope is exactly the average velocity over the interval. As the second point moves toward the first, the secant line approaches the tangent line when the curve has a well-defined tangent there. The limiting secant slope is therefore the tangent slope.

This connection explains why a tangent drawn on a graph can be used to estimate instantaneous velocity. It also explains why simply choosing two points on the curve and calculating their secant slope gives an average velocity, not automatically an instantaneous one.

Deriving the Tangent-Slope Connection

The relationship follows directly from the definition of derivative. The proof also identifies exactly what is being estimated when nearby position values are used.

The condition that the derivative exists matters. If the left-hand and right-hand difference quotients approach different values, there is no single derivative value at that instant, and the graph does not have one tangent slope in the derivative sense. A tangent estimate is meaningful when the graph is sufficiently smooth near the point and its tangent direction is clear.

Worked Example: Finding Instantaneous Velocity from a Position Formula

Estimating Velocity from Nearby Position Data

In some problems, position is provided in a table rather than by a formula. A natural estimate uses two nearby values on opposite sides of the time of interest. This is a centered difference quotient: it approximates the tangent slope using a short secant through points on both sides. It is still an estimate unless the problem provides enough information to determine the exact derivative.

Worked Example: Estimating from a Table

For actual measured data, the exact derivative is usually unknown, and position measurements may be rounded or noisy. A centered estimate can be useful, but it does not turn approximate data into an exact velocity. The time spacing and precision of the given values affect how reliable the estimate is.

Estimating Velocity from a Position-Time Graph

To estimate a tangent slope from a graph, first identify the time of interest and the point on the position curve. Then use the drawn tangent line, not an arbitrary line through two curve points. Choose two well-separated points on that tangent line, read their coordinates using the axis scales, and calculate rise divided by run. Using points farther apart on the tangent can reduce errors from reading a very small change, but the points must lie on the tangent line.

Worked Example: Reading a Tangent Slope from a Graph

Average Velocity Is Not Instantaneous Velocity

A common error is to calculate an average velocity over a noticeable interval and report it as the instantaneous velocity at one endpoint. The two quantities may happen to match in a particular situation, but that agreement should not be assumed. The interval average uses two distinct times; the instantaneous value requires a limit or a tangent slope at the specified time.

The correction is to match the method to the question. For a finite time interval, calculate average velocity. For one specified instant, use the derivative definition, a derivative value supplied by the problem, or a tangent-slope estimate from the graph.

When the Instantaneous Velocity Does Not Exist

The derivative-based definition requires the difference quotient to approach one finite value. A position function can fail that test even when the position itself is defined. For example, consider \(s(t)=|t|\) at \(t=0\). For \(h>0\), the quotient is \(\frac{|h|-|0|}{h}=1\). For \(h<0\), it is \(\frac{|h|-|0|}{h}=-1\). The one-sided limits disagree, so the two-sided derivative does not exist at \(0\). There is no single derivative-based instantaneous velocity there.

This is why graph estimates require care near a sharp corner or another point where the curve does not have a unique tangent direction. A graph can show a position at a time without having a well-defined tangent slope at that time.

What to Keep in Mind

Instantaneous velocity links three descriptions of the same rate: a limit of average velocities, the derivative of position, and the tangent slope on a position-time graph. In calculations, keep track of which one is exact and which one is estimated. A derivative found from a formula can be exact; a slope read from a graph or estimated from a table is generally approximate.

Always include units and identify the time at which the velocity is being reported. For position measured in meters and time in seconds, a velocity is expressed in meters per second. If the calculation uses two separated times, label it an average velocity over that interval. If the question asks for an instant, explain how the derivative or tangent slope provides the instantaneous value.