Calculus I · 2B · lesson
Speeding Up and Slowing Down
Learn speeding up and slowing down with clear exposition, guided derivations, worked examples, visuals, checks, and interpretation.
Section overview
Interpretation, motion, and ratesWhat this section is building
Learn speeding up and slowing down with clear exposition, guided derivations, worked examples, visuals, checks, and interpretation.
Position, velocity, and acceleration are synchronized views: amount, rate of amount, and rate of the rate.
Separate direction from speed, and compare the signs of velocity and acceleration before describing speeding behavior.
Treating negative velocity as slowing down or confusing a function's height with the slope of its graph.
Learning objectives
Determine when a particle speeds up or slows down.
Compare the Signs of Velocity and Acceleration
Before the formulas
The graphs in Speeding Up and Slowing Down should be read vertically through a common input. A height on the derivative graph is a slope on the original graph. A zero of velocity is a horizontal tangent of position; a zero of acceleration is a horizontal tangent of velocity.
Do not match graphs by superficial shape. Translate one feature at a time: sign, zeros, increasing behavior, and concavity. This produces a defensible interpretation even when the graphs are unfamiliar or not drawn to a convenient scale.
Read this graph as text
Speeding up and slowing down depend on two signs. An object speeds up when velocity and acceleration have the same sign and slows down when their signs differ. The rule follows from whether the magnitude |v| is moving away from or toward zero. Negative velocity does not mean slowing down. It means motion in the negative direction. If acceleration is also negative, the velocity becomes more negative and the speed increases. The table organizes all four sign combinations so you can reason rather than guess from the word "negative."
Every relationship in speeding up and slowing down depend on two signs is identified with written labels plus distinct solid, dashed, dotted, double, marker, or pattern cues; color is never the only carrier of meaning.
Why it matters: This matrix targets one of the most persistent motion misconceptions: confusing negative velocity with deceleration. The key idea is magnitude. The labels "more negative" and "toward zero" make that mechanism explicit.
An object speeds up when velocity and acceleration have the same sign and slows down when their signs differ. The rule follows from whether the magnitude |v| is moving away from or toward zero.
Speed changes according to whether velocity and acceleration cooperate
Speed is , so it increases when acceleration pushes velocity farther from zero. That occurs when velocity and acceleration have the same sign. Speed decreases when acceleration pushes velocity toward zero, which occurs when their signs differ.
A sign chart is safer than verbal intuition here. Negative velocity means motion in the negative direction, not automatically slowing down. An object with and is moving left faster and faster.
Speed is the magnitude , so an object speeds up when velocity moves away from zero and slows down when velocity moves toward zero. The clean sign test follows: velocity and acceleration with the same sign imply speeding up; opposite signs imply slowing down.
This is easier to understand on a number line than through memorization. Acceleration points in the direction that velocity is being pushed. If that push reinforces the current direction, speed grows; if it opposes the current direction, speed shrinks.
Sign test for speed
• If and are both positive, the particle moves forward faster. • If and are both negative, the particle moves backward faster. • If their signs differ, the particle slows down.
Use a combined sign chart
For the previous motion
determine speeding-up and slowing-down intervals on .
Worked solution
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motion-sign-01At an instant, m/s and m/s. Is the object speeding up or slowing down?
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Speed increases when velocity and acceleration have the same sign.
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