Your dashboard says 60 km/h. It looks like a measurement made at this very instant.
But the familiar distance-over-time calculation needs an interval. How can an instrument describe your speed right now?
The answer begins with a distinction: mathematics defines velocity at an instant, while an instrument estimates it from signals collected over time.
In How Can a Curve Have a Slope at Just One Point?, a secant line became a tangent as two points moved closer. Put time on the horizontal axis, and that same idea becomes a way to describe motion.
One photograph tells you where, not how fast
Imagine a ball on a straight track. Pause a recording and mark its location: 10.5 meters from the start. Is the ball moving at 2 m/s, 5 m/s, or not moving at all?
A single position is not enough to tell. You need information about nearby moments, or a physical model that connects position with time.
Two video frames separated by a known time give a first useful quantity: average velocity, the change in position divided by the elapsed time.
average velocity = change in position / elapsed time
When the ball moves along one straight coordinate axis, we can write this as:
v̄ = [x(t + h) − x(t)] / h
Here x(t) is its position at time t, while h is a nonzero time interval. The result describes an interval, not an exact instant.
Follow an accelerating object for one second, then less
Consider an idealized object moving in a straight line, with position in meters given by
x(t) = 2t + 0.5t²
Here t is the numerical time in seconds; the coefficients carry the necessary units: 2 m/s and 0.5 m/s².
At t = 3 seconds, its position is 10.5 meters. What is its velocity at that moment?
Begin with a later measurement at 4 seconds. The object is at 16 meters, so its average velocity from 3 to 4 seconds is (16 − 10.5) / 1 = 5.5 m/s.
But that number includes an entire second during which the object keeps gaining speed.
Now move the later measurement closer to 3 seconds.
| Time interval after 3 s | Average velocity |
|---|---|
| 1.00 s | 5.500 m/s |
| 0.50 s | 5.250 m/s |
| 0.10 s | 5.050 m/s |
| 0.01 s | 5.005 m/s |
The numbers approach 5 m/s. Intervals chosen just before 3 seconds approach the same value from below.
Both sides agree, so we have a well-defined local rate of change.
What happens when the interval reaches zero?
We never divide by zero. Instead, we ask what number the ratio approaches as h gets closer and closer to zero.
With the position formula above, the average velocity beginning at 3 seconds simplifies to:
[x(3 + h) − x(3)] / h = 5 + 0.5h, for h ≠ 0.
As h approaches zero, this expression approaches 5.
This is a limit: the value approached by nearby calculations, not the result of substituting zero into a forbidden denominator.
In calculus, the result is called instantaneous velocity. More generally,
v(t) = limh→0 [x(t + h) − x(t)] / h = dx/dt
The symbol dx/dt means the local rate at which position changes with time. On a position–time graph, it is the slope of the tangent line.
The geometry of a curve has become a physical tool: its local slope tells us the velocity.
Velocity has a direction. Speed does not.
Suppose the object turns around and travels in the negative direction. Its instantaneous velocity might become −5 m/s.
The minus sign tells us the direction along the chosen axis; it does not mean the object has “negative speed.”
Speed is the magnitude of velocity. For motion along one line, a velocity of either +5 m/s or −5 m/s corresponds to a speed of 5 m/s.
A car’s dashboard normally displays speed, not the signed velocity along your coordinate axis.
In the accelerating example above, motion is always in the positive direction, so the numerical values of velocity and speed agree. For motion that reverses direction, their distinction matters.
How does the speedometer get its number?
Modern vehicles can use rotational information from wheel or drivetrain sensors. A wheel-speed sensor detects repeating changes as an encoder rotates.
The pulse frequency relates to rotational speed, and electronic systems use those signals to estimate vehicle speed.
Implementations differ by vehicle. Wheel slip and other conditions can affect the relationship between wheel rotation and true ground speed.
That reading is an estimate of motion now, assembled from information acquired over a finite interval.
It is not a direct measurement in a literally zero-length interval. The mathematical derivative gives the ideal quantity the estimate aims to represent.
Try it: can an interval be too short?
Return to x(t) = 2t + 0.5t². Predict what happens if you reduce the time step from 1 second to 0.1 second to 0.01 second.
The table suggests that the ideal estimate gets closer to 5 m/s.
Real positions are measured imperfectly. Suppose two recorded positions each have an error no larger than 0.01 m.
Their difference could then have an error as large as 0.02 m. Divide by 0.1 second, and that error can contribute up to 0.2 m/s to the estimated velocity.
Divide by 0.01 second, and the worst-case contribution rises to 2 m/s.
So shorter is not automatically better.
This is a first taste of numerical differentiation: when a computer estimates change from discrete measurements, it must balance approximation error against measurement error.
For a quick calculation, try this optional Python example:
def position(t):
return 2*t + 0.5*t*t
t = 3.0
for h in (1.0, 0.5, 0.1, 0.01):
average = (position(t+h) - position(t))/h
print(f"h={h:4.2f} s velocity={average:6.3f} m/s")This code uses an ideal mathematical position function, so it has no sensor noise. Real measured data require additional care.
From the instant to what happens next
The speedometer’s reading and the tangent’s slope describe the same underlying idea from two directions.
Mathematics defines the exact local rate of change; instruments and computation seek a reliable estimate of it.
The number will not always stay at 5 m/s. If velocity itself changes, what measures that change?
That question leads to acceleration—and to another derivative.
Words to Keep
- Average velocity: displacement divided by elapsed time.
- Instantaneous velocity: the derivative of position with respect to time.
- Speed: the magnitude of velocity.
- Numerical differentiation: estimating a rate of change using discrete numerical values.