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Calculus I · Unit 2B · lesson

Speeding Up and Slowing Down

Concept

Learning objectives

Determine when a particle speeds up or slows down.

Compare the Signs of Velocity and Acceleration

Explanation

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.

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.
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.

Visual study

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.

Explanation

Speed changes according to whether velocity and acceleration cooperate

Speed is ∣v∣|v|, 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 v<0v<0 and a<0a<0 is moving left faster and faster.

Speed is the magnitude ∣v∣|v|, 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.

Method

Sign test for speed

• If vv and aa are both positive, the particle moves forward faster. • If vv and aa are both negative, the particle moves backward faster. • If their signs differ, the particle slows down.

Guided walkthrough

Use a combined sign chart

For the previous motion

v(t)=3(t−1)(t−3),a(t)=6(t−2),v(t)=3(t-1)(t-3),\qquad a(t)=6(t-2),

determine speeding-up and slowing-down intervals on (0,5)(0,5).

Answer reveal

Worked solution

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Interactive checkmotion-sign-01

At an instant, v=−4v=-4 m/s and a=−2a=-2 m/s2^2. Is the object speeding up or slowing down?

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Show hint

Speed increases when velocity and acceleration have the same sign.

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After the explanation

Use the section idea

Reading lens

Use the sign, size, units, and zeros of derivatives to tell a time-aligned story about motion or another changing quantity.

Mental model

Position, velocity, and acceleration are synchronized views: amount, rate of amount, and rate of the rate.

Decision

Separate direction from speed, and compare the signs of velocity and acceleration before describing speeding behavior.

Common trap

Treating negative velocity as slowing down or confusing a function's height with the slope of its graph.

Check yourself

Can you interpret a first and second derivative at the same input without mixing their units or meanings?

Source & rights

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