Lesson 6 of 6 · Average rates, acceleration & an optional derivative
Read motion as a changing relationship
9 minute lesson · 3 checks along the way
Your goal for this lessonCalculate a signed average rate, distinguish it from an instantaneous value, and recognize the derivative as an optional extension of a familiar slope.
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Take a change over an interval
A rate compares a change in one quantity with an elapsed interval. In a fictional docking exercise, range to a target falls from 120 meters to 100 meters over ten seconds. The average is (100 − 120) m / 10 s = −2 m/s. The negative sign means decreasing range under our convention. It does not mean the elapsed time is negative or that the spacecraft has a negative speed.
Range rate describes approach or recession along the changing line of sight. It does not by itself specify the full velocity ; sideways motion can also be present. The ISS Progress 63 photograph records one viewing moment, so it cannot supply the exercise’s two ranges and times. A motion claim needs measurements across time, their reference frame, and an account of relevant measurement errors.
This NASA photograph records Progress MS-02, the ISS Progress 63 mission, approaching the International Space Station. It supplies context for relative motion. The 120-meter and 100-meter ranges and ten-second interval are invented lesson inputs, not measurements extracted from this photograph.
Look closely
Can the spacecraft’s apparent size in this one frame establish its closing rate?
See what to notice
No. A range rate requires a change in range over a known time interval, with a defined observer and measurement method. A single frame does not separate motion from distance, camera field, or image scaling.
Check 1 of 3
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Separate a local value from an average
The ten-second average does not imply a constant rate throughout the interval. A vehicle might close rapidly at first and slow near the end. On a linear range-versus-time graph, joining the two endpoints gives a whose slope is the interval’s average rate. Measuring over shorter intervals can reveal changing behavior, provided the timing and range measurements are precise enough.
Acceleration describes a change in velocity per time. For one-dimensional motion along a fixed axis, suppose velocity changes from −2 m/s to −1 m/s over five seconds. The is [−1 − (−2)]/5 = +0.20 m/s². The object still moves in the negative direction, but its speed decreases. A positive acceleration does not always mean speeding up; its relation to velocity matters.
Check 2 of 3
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Work through a decision
Estimate what happens near one instant
An optional bridge uses a smooth invented position model: x(t) = (0.25 m/s²) × t². At four seconds, x is four meters; at six seconds, it is nine meters. The average velocity from four to six seconds is (9 − 4)/2 = 2.5 m/s. At 4.2 seconds, x is 4.41 meters, giving 2.05 m/s over the shorter interval starting at four seconds.
As these forward intervals shrink toward zero, the calculated average approaches 2 m/s. For this smooth model, that limiting slope is the instantaneous velocity at four seconds. Calculus calls it a and writes dx/dt. You do not calculate it by dividing a change of zero by zero. You investigate the value approached by ratios over nonzero intervals; the limit must exist.
Optional: from interval slope to local slope
The curve is an explicitly invented mathematical model, not a fit to mission telemetry. Formal derivative rules are optional; the ordinary checks use interval changes and units.
Open full-size diagram in a new tab: Optional: from interval slope to local slopeKeep the extension optional and the evidence real
You can complete the lesson using changes, intervals, and units without learning derivative rules. The optional notation connects a familiar graph slope with a more general mathematical tool. Not every graph has a well-defined instantaneous slope at every point. A sharp corner in an idealized model needs special care, and a few noisy measurements do not establish an exact smooth curve between them.
Smaller measurement intervals also make fixed errors a larger fraction of the difference being measured. Real analysts choose suitable sampling and models rather than assuming that ever-shorter intervals automatically produce perfect velocities. Report whether a value is an interval average, a model’s instantaneous prediction, or an estimate from observations. This distinction lets a useful mathematical result retain its connection to what was actually measured.
Check 3 of 3
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Put it together
Your lesson progress
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Go to the source
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