From Sampled Positions to Velocity
Position data usually come as a table of measurements at selected times. Such a table tells us where an object was at those times, but it does not directly report its instantaneous velocity at any one of them. We can estimate that velocity by comparing nearby position readings.
The basic calculation is a change in position divided by a change in time. The choice of which readings to use matters: readings farther apart describe motion over a longer interval, while readings closer to the target time are more local. Measurement precision matters too, especially when the position difference is small.
If position is measured in metres and time in seconds, both average and instantaneous velocity have units of metres per second, written \(\text{m/s}\). A table-based calculation gives an estimate, not automatically the exact value of \(s'(t)\).
Using Readings on Both Sides
Suppose the target time is \(t\), and the table includes readings at \(t-h\) and \(t+h\), where \(h>0\). The centered difference estimate is the change in position across those two readings divided by the full time interval \(2h\):
This estimate uses the same time spacing on each side of the target. The numerator is the change from the earlier reading to the later reading, and the denominator is the elapsed time between those readings—not the spacing \(h\) on just one side.
Worked Example: Estimating Velocity from Symmetric Readings
A position table gives \(s(1.8)=6.84\text{ m}\) and \(s(2.2)=9.24\text{ m}\). Estimate the instantaneous velocity at \(t=2.0\text{ s}\) using these readings.
The readings are equally spaced around \(2.0\text{ s}\), with \(h=0.2\text{ s}\). The full interval is \(2h=0.4\text{ s}\). Therefore,
The table-based estimate at \(2.0\text{ s}\) is \(6.0\text{ m/s}\). The units check: metres divided by seconds gives metres per second. This calculation is the average velocity over the interval from \(1.8\text{ s}\) to \(2.2\text{ s}\); it estimates, rather than directly reads off, the instantaneous velocity at the middle time.
Why the Centered Difference Estimates the Derivative
The derivative is defined through nearby changes in position. To see why using readings on both sides works, split the centered difference into two one-sided quotients. For \(h>0\),
The first quotient on the right is a forward difference. The second is a backward difference, written with positive time spacing \(h\). If \(s\) is differentiable at \(t\), each quotient approaches \(s'(t)\) as \(h\) approaches zero. Their average therefore also approaches \(s'(t)\).
This is a limiting statement: it describes what happens as the spacing becomes arbitrarily small for exact values of \(s\). A real table has a finite spacing and measured values, so its centered difference is an approximation. A smaller \(h\) is not automatically better when the entries have been rounded or measured with error.
Comparing Estimates at Different Spacings
When readings are available at several distances from the target time, calculate estimates at more than one spacing. If the estimates settle toward a common value as the spacing shrinks, that supports the use of that value as a local estimate. The comparison is informative, but a finite table alone cannot guarantee that the limiting derivative has been found.
Worked Example: Comparing Two Time Spacings
Suppose position readings, in metres, are shown below. Estimate the instantaneous velocity at \(t=1\text{ s}\) using \(h=0.2\text{ s}\) and \(h=0.1\text{ s}\).
| Time \(t\) (s) | Position \(s(t)\) (m) |
|---|---|
| 0.8 | 0.512 |
| 0.9 | 0.729 |
| 1.0 | 1.000 |
| 1.1 | 1.331 |
| 1.2 | 1.728 |
For \(h=0.2\text{ s}\), use the readings at \(0.8\text{ s}\) and \(1.2\text{ s}\). The full interval is \(0.4\text{ s}\):
For \(h=0.1\text{ s}\), use the readings at \(0.9\text{ s}\) and \(1.1\text{ s}\). The full interval is \(0.2\text{ s}\):
The two estimates are \(3.04\text{ m/s}\) and \(3.01\text{ m/s}\). They are close, and the estimate with the smaller spacing is closer to \(3.00\text{ m/s}\). For a check, these table entries agree with \(s(t)=t^3\), whose derivative is \(s'(t)=3t^2\); at \(t=1\), this gives \(s'(1)=3\text{ m/s}\). In an actual measurement problem, the underlying formula may not be known, so the comparison of estimates is a practical consistency check, not a proof of the exact velocity.
Measurement Precision and the Choice of Spacing
Suppose each recorded position differs from its true value by at most \(\delta\) metres. In a centered difference, there are two recorded positions. Each can contribute an error of magnitude at most \(\delta\), so the error in their difference is at most \(2\delta\) metres. Dividing by the full time interval \(2h\) gives the bound
This bound concerns the effect of position measurement error alone; it does not include the difference between a finite-spacing estimate and the true instantaneous velocity. It shows why a very small spacing can be troublesome: the same position uncertainty is divided by a smaller time interval.
Worked Example: Assessing the Effect of Rounded Positions
Suppose positions are recorded to the nearest \(0.01\text{ m}\), so each recorded value differs from the true position by at most \(\delta=0.01\text{ m}\). Compare the measurement-error bounds for centered estimates with \(h=0.1\text{ s}\) and \(h=0.5\text{ s}\).
For \(h=0.1\text{ s}\), the full interval is \(0.2\text{ s}\). The maximum error contribution from the two position readings is \(0.02\text{ m}\), so
For \(h=0.5\text{ s}\), the full interval is \(1.0\text{ s}\), giving
The larger spacing has a smaller bound on the measurement-error contribution. However, it averages motion across a wider time interval, so it may be less representative of the instantaneous velocity at the target time if velocity changes substantially over that interval. The calculations quantify one source of uncertainty; they do not establish which estimate is closer to the instantaneous value without more information about the motion.
There is a balance to consider. A wide interval may blur changes in velocity because it describes motion over a longer period. A narrow interval is more local, but rounding or measurement noise can have a larger effect on the quotient. In practice, use the available precision, compare estimates at several spacings when possible, and state clearly that the result is an estimate.
A Common Pitfall: Using Only One Side and Calling It Exact
A forward difference uses the target reading and one later reading. It estimates the derivative, but it is not generally equal to the instantaneous velocity at the target time. For example, let \(s(t)=t^2\) metres, with \(t\) measured in seconds, and estimate at \(t=2\text{ s}\) using \(h=0.5\text{ s}\). The forward quotient is
Calling \(4.5\text{ m/s}\) the exact instantaneous velocity at \(2\text{ s}\) is the mistake: it is an average velocity from \(2\text{ s}\) to \(2.5\text{ s}\). For this example, \(s'(t)=2t\), so \(s'(2)=4\text{ m/s}\). The centered difference at the same spacing gives
Here the centered estimate matches the derivative exactly. That match is a feature of this particular function and spacing, not a guarantee for every position table. The safe practice is to label a table calculation as an estimate and identify the interval used.
How to Report a Table-Based Estimate
A clear report specifies the target time, the readings used, the calculation, and the units. It also distinguishes the estimate from an exact derivative value.
Identify the table entries before and after the time where velocity is to be estimated.
For a centered difference, confirm that the two readings are equally spaced around the target; use their full time separation in the denominator.
Subtract the earlier position from the later position, divide by elapsed time, and include position-per-time units.
Compare estimates at different spacings if possible, and note whether measurement rounding could significantly affect the quotient.
Position tables turn the derivative into a practical estimation problem. The difference quotient supplies the calculation, the derivative definition explains why nearby quotients can approximate instantaneous velocity, and the table's spacing and precision indicate how cautiously the result should be interpreted.
Check Your Understanding
Use the definitions and methods in this tutorial to answer the following questions.
- A position changes from \(8.0\text{ m}\) at \(t=3\text{ s}\) to \(14.0\text{ m}\) at \(t=5\text{ s}\). What is the average velocity over this interval, including units?
- At \(t=4\text{ s}\), a table has readings at \(3.8\text{ s}\) and \(4.2\text{ s}\). What time interval belongs in the denominator of the centered difference?
- Why does the centered difference approach the derivative when \(h\) approaches zero and the position function is differentiable at the target time?
- If each of two position readings has uncertainty at most \(0.02\text{ m}\), what is the resulting measurement-error bound for a centered estimate with \(h=0.2\text{ s}\)?
- Why can decreasing the spacing make an estimate more local but also make position rounding more influential?
