Straight-Line Motion: Speeding Up and Slowing Down

The Three Quantities You Need

In straight-line motion, an object's position, velocity, and acceleration are connected by derivatives. If represents position as a function of time, then velocity is the first derivative of position and acceleration is the second derivative.

The meanings are important:

• Position tells you where the object is.
• Velocity tells you the direction in which the object is moving and how quickly its position is changing.
• Acceleration tells you how velocity is changing.

For a straight-line motion problem, the sign of velocity tells you the direction of motion, while the sign of acceleration tells you the direction in which velocity is changing.

The Big Idea: Compare the Signs

The most important rule in this entire topic is that speeding up and slowing down are determined by the relationship between velocity and acceleration, not by whether either quantity is positive or negative by itself.

Why does this work? Think about what acceleration does to velocity.

If velocity is positive and acceleration is positive, the velocity becomes more positive. The object is moving in the positive direction and becoming faster.

If velocity is negative and acceleration is negative, the velocity becomes more negative. The object is moving in the negative direction and its speed is increasing.

But if velocity and acceleration have opposite signs, acceleration pushes the velocity toward zero. The object's speed decreases, so it is slowing down.

The Four Sign Combinations

Velocity	Acceleration	What Happens?
Positive	Positive	Speeding up
Positive	Negative	Slowing down
Negative	Positive	Slowing down
Negative	Negative	Speeding up

A useful memory trick is:

In other words, same signs = speeding up and opposite signs = slowing down.

Why Negative Velocity Does Not Mean Slowing Down

This is one of the most common mistakes on AP Calculus questions.

Suppose an object has velocity . The negative sign means the object is moving in the negative direction. It does not mean the object is slowing down.

If the acceleration is also negative, such as , then velocity and acceleration have the same sign. The object's speed is increasing.

For example, imagine a car traveling backward. If the car is moving backward faster and faster, its velocity is negative and its acceleration is negative. It is clearly speeding up.

A Step-by-Step Method

1. Find the velocity. If you are given position , calculate .
2. Find the acceleration. Calculate .
3. Identify the interval or time values where the question asks about the object's motion.
4. Determine the sign of velocity and acceleration on each interval.
5. Compare the signs. Same signs mean speeding up; opposite signs mean slowing down.

Worked Examples

A Faster AP Exam Method

On an AP Calculus question, you usually do not need to calculate the actual speed. You only need to determine whether velocity and acceleration have the same or opposite signs.

For a quick solution, make a tiny sign chart:

Then interpret each row:

• Same signs → speeding up.
• Opposite signs → slowing down.

This is especially useful when the problem gives you graphs of and rather than explicit formulas.

Using Graphs

You can determine speeding up and slowing down directly from a velocity graph if you understand what acceleration means.

Acceleration is the derivative of velocity:

Therefore, acceleration is the slope of the velocity graph.

To analyze a velocity graph:

1. Determine whether the velocity graph is above or below the horizontal axis. This tells you the sign of velocity.
2. Determine whether the velocity graph is increasing or decreasing. This tells you the sign of acceleration.
3. Compare the two signs.

What Happens When Velocity Is Zero?

If , the object's instantaneous speed is zero. At that exact instant, the object is neither speeding up nor slowing down in the usual AP Calculus classification because its speed is zero.

However, this does not mean that acceleration must also be zero.

For example, at the top of a vertically thrown object's path, velocity is zero for an instant while acceleration remains negative because gravity continues to act on the object.

So do not conclude that an object has zero acceleration simply because its velocity is zero.

Speed Versus Velocity

This distinction is essential.

Velocity includes direction and can be positive or negative. Speed is the magnitude of velocity and is never negative.

Because speed is the absolute value of velocity, an object can have a negative velocity while its speed is increasing.

For example, if velocity changes from to , the velocity becomes more negative, but the speed changes from to . The object is therefore speeding up.

Common Mistakes

The Essential Rule to Memorize

That's the entire idea. The challenge on AP Calculus is usually not the rule itself; it is correctly determining the signs of velocity and acceleration from formulas, tables, graphs, or verbal descriptions.

Practice Problems