AP Calculus AB 4.2 Rectilinear Motion: Finding Velocity, Acceleration, at Rest, and Displacement

In AP Calculus AB, Section 4.2 focuses on rectilinear motion, emphasizing how to find velocity, acceleration, conditions of rest, and displacement from a position function that describes the motion of an object along a straight line. This section explores the relationships between these concepts and how they apply to real-world situations involving motion. Here’s a detailed overview of the key concepts covered:

### Key Concepts:

1. **Position Function**:
- The position of an object moving along a straight line is represented by a function \( s(t) \), where \( s \) denotes the position (in units such as meters) and \( t \) represents time (in seconds).
- This function provides the location of the object at any given moment and is the foundation for calculating velocity and acceleration.

2. **Velocity**:
- Velocity is defined as the rate of change of position with respect to time and is mathematically represented as the first derivative of the position function:
\[
v(t) = s'(t)
\]
- The velocity function \( v(t) \) indicates how fast and in what direction the object is moving at time \( t \).
- Positive values of \( v(t) \) indicate motion in the positive direction, while negative values indicate motion in the negative direction.

3. **Acceleration**:
- Acceleration is the rate of change of velocity with respect to time, expressed as the derivative of the velocity function or the second derivative of the position function:
\[
a(t) = v'(t) = s''(t)
\]
- The acceleration function \( a(t) \) describes how the object's velocity is changing over time. Positive acceleration signifies an increase in velocity, whereas negative acceleration (deceleration) indicates a decrease in velocity.

4. **Finding When the Object is At Rest**:
- An object is considered to be at rest when its velocity is zero. To find the times when the object is at rest, students solve the equation:
\[
v(t) = s'(t) = 0
\]
- This involves finding the critical points of the velocity function, which can indicate where the object changes direction or stops moving.

5. **Displacement**:
- Displacement refers to the change in position of an object over a given time interval. It is calculated as:
\[
\text{Displacement} = s(b) - s(a)
\]
where \( s(b) \) is the position at the end of the interval and \( s(a) \) is the position at the beginning of the interval.
- Displacement can be positive, negative, or zero, depending on the direction of motion.

6. **Example Problem**:
- For a position function such as \( s(t) = t^3 - 6t^2 + 9t \):
- **Velocity**: Calculate \( v(t) = s'(t) = 3t^2 - 12t + 9 \).
- **Acceleration**: Calculate \( a(t) = v'(t) = s''(t) = 6t - 12 \).
- **At Rest**: Set \( v(t) = 0 \) and solve for \( t \) to find when the object is at rest.
- **Displacement**: Evaluate \( s(b) - s(a) \) over a specified interval to determine the overall change in position.

### Applications:
- The concepts of rectilinear motion are essential in fields such as physics, engineering, and kinematics, where the analysis of motion along a straight path is crucial.
- Students apply these principles to solve real-world problems involving vehicles, projectiles, and other moving objects, enhancing their understanding of motion dynamics.

### Summary:
Section 4.2 of AP Calculus AB provides a comprehensive understanding of rectilinear motion by examining the relationships between position, velocity, acceleration, conditions of rest, and displacement. By mastering these concepts, students develop essential skills for analyzing and interpreting motion mathematically, which lays the groundwork for more advanced topics in calculus and its applications in the real world.

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Nick Perich
Norristown Area High School
Norristown Area School District
Norristown, Pa

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