Introduction to Straight-Line Motion

Welcome to one of the most practical chapters in AP Calculus BC! Have you ever wondered how a self-driving car knows exactly when to brake, or how a physicist calculates the path of a particle? It all comes down to Straight-Line Motion (also called rectilinear motion). In this chapter, we use derivatives to understand the relationship between where an object is, how fast it is moving, and how its motion is changing.

Don't worry if physics isn't your favorite subject. In Unit 4, we focus specifically on the contextual application of differentiation—essentially, how to use the derivative rules you already know to solve real-world "moving" problems.

The Hierarchy of Motion

In calculus, we track a particle moving along a straight line (usually the \(x\)-axis or \(y\)-axis) using three main functions of time, \(t\). These functions are linked by the derivative.

1. Position: \(s(t)\) or \(x(t)\)

The position function tells you exactly where the particle is at a specific time \(t\). It is a coordinate on a line.

  • If \(s(t) > 0\), the particle is to the right of the origin (or above it).
  • If \(s(t) < 0\), the particle is to the left of the origin (or below it).
  • If \(s(t) = 0\), the particle is at the origin.

2. Velocity: \(v(t)\)

Velocity is the instantaneous rate of change of position with respect to time. In other words, it is the derivative of position.

\(v(t) = s'(t) = \frac{ds}{dt}\)

Velocity is a "vector" quantity in physics, which in our straight-line world just means the sign matters:

  • Positive velocity (\(v(t) > 0\)): The particle is moving in the positive direction (right or up).
  • Negative velocity (\(v(t) < 0\)): The particle is moving in the negative direction (left or down).
  • Zero velocity (\(v(t) = 0\)): The particle is momentarily "at rest."

3. Acceleration: \(a(t)\)

Acceleration is the rate of change of velocity. It is the derivative of velocity, which makes it the second derivative of position.

\(a(t) = v'(t) = s''(t) = \frac{dv}{dt}\)

Acceleration tells us how the velocity is changing. If \(a(t) > 0\), the velocity is increasing; if \(a(t) < 0\), the velocity is decreasing.

Key Takeaway: To move "down" the hierarchy (Position \(\to\) Velocity \(\to\) Acceleration), just take the derivative!

Speed vs. Velocity

Students often use these words interchangeably in daily life, but AP Calculus makes a strict distinction. Speed is the magnitude of velocity. It does not care about direction.

\(\text{Speed} = |v(t)|\)

Because speed is an absolute value, it can never be negative. If your velocity is \(-55\) mph, your speed is simply \(55\) mph.

The "Speeding Up" vs. "Slowing Down" Trap

One of the most common questions on the AP Exam asks if a particle is speeding up or slowing down at a specific time. Warning: You cannot look at acceleration alone to answer this!

Think of it this way:

  • Speeding Up: Velocity and acceleration have the same sign.
    Example: You are moving right (\(v > 0\)) and being pushed right (\(a > 0\)). You go faster.
    Example: You are moving left (\(v < 0\)) and being pushed left (\(a < 0\)). You go faster in the negative direction.

  • Slowing Down: Velocity and acceleration have opposite signs.
    Example: You are moving right (\(v > 0\)) but being pulled left (\(a < 0\)). This is like hitting the brakes.

Quick Rule of Thumb:
If \(v(t) \cdot a(t) > 0\), the particle is speeding up.
If \(v(t) \cdot a(t) < 0\), the particle is slowing down.

Interpreting Direction Changes

A particle changes direction when its velocity changes from positive to negative, or vice versa. To find these moments:

  1. Set \(v(t) = 0\) and solve for \(t\).
  2. Check if the sign of \(v(t)\) actually changes at those points (using a sign chart or a graph).

Note: Just because \(v(t) = 0\) doesn't mean it changed direction. It could have paused and then continued in the same direction!

Units and Notation

On the AP Exam, especially in the Free-Response Section (FRQ), units are mandatory if the question provides them. If position is in meters (\(m\)) and time is in seconds (\(s\)):

  • Position: \(m\)
  • Velocity: \(m/s\) (meters per second)
  • Acceleration: \(m/s^2\) (meters per second squared)

Did you know? Acceleration units \(m/s^2\) literally mean "\(meters \text{ per } second, \text{ per } second.\)" It tells you how many units of velocity are added every second.

Summary Checklist for Students

  • Velocity is the derivative of Position.
  • Acceleration is the derivative of Velocity.
  • Speed is the absolute value of Velocity.
  • To check if an object is speeding up, check if \(v(t)\) and \(a(t)\) have the same sign.
  • To find when an object is at rest, solve \(v(t) = 0\).
  • Always include units in your final answer when context is given (e.g., \(ft/sec\)).

Cross-reference: While this chapter focuses on using derivatives to find motion values, you will later use integrals in Unit 8: Applications of Integration to go backward from acceleration to velocity and position.

Calculator Usage Note

For the calculator-active sections of the AP Exam (Section I Part B and Section II Part A), you are expected to use your graphing calculator to:

  • Calculate the numerical derivative of a position function to find velocity at a point.
  • Find the zeros of a velocity function to determine when a particle is at rest.

When doing this on an FRQ, always write the setup, such as \(v(3) = s'(3)\), before writing the numerical value from your calculator!