Solving Motion Problems Using Parametric and Vector-Valued Functions

#9.6 Solving Motion Problems Using Parametric and Vector-Valued Functions

In this study guide, we are going to apply what you’ve learned about vector-valued and parametric functions to motion problems so you can take on anything the AP Calculus BC exam throws your way. From understanding the nuances of displacement and distance traveled to integrating vector-valued functions, you will develop the skills needed to solve complex motion problems. 🚗

#🚘 Recap of Motion Terms

Before we dive into the problems, let’s refresh our memory on the definitions of position, velocity, acceleration, displacement, and distance traveled. If you want a more thorough review than the one provided below, we’ve got you covered.

#📍Position

Position refers to the location of an object at any given time. It is a measure of where the object is situated within a coordinate system. In calculus, we denote a position function using $s(t)$.

#🚄Velocity

Velocity is the rate of change of position with respect to time. In more straightforward terms, it tells us how fast an object is moving and in what direction. Mathematically, velocity is the derivative of position, denoted as $v(t) = \frac{ds}{dt}$, where $\frac{ds}{dt}$ represents the rate of change of position.

#🚤Acceleration

Acceleration is the rate of change of velocity with respect to time. It describes how quickly the velocity of an object is changing. In mathematical terms, acceleration is the derivative of velocity, denoted as $a(t)=\frac{dv}{dt}$, where $\frac{dv}{dt}$ represents the rate of change of velocity.

Since acceleration is the derivative of the velocity, and the velocity is the derivative of the position, acceleration can also be defined as the second derivative of the position:

$$a(t)=\frac{d^2s}{dt^2}$$

#↔️Displacement

Displacement is a straightforward measure of the change in position of an object. It tells us how far the object has moved from its initial position to its final position, regardless of the path it took to get to its final position. Mathematically, it is represented as the following, where $\Delta s$ is the displacement, and $s_{final}$ and $s_{initial}$ are the final and initial positions, respectively.

$$\Delta s = s_{final}-s_{initial}$$

If you want to know how much ground an object has covered in total, look no further than its displacement.

!Untitled

Two different paths: distance versus displacement.

Image Courtesy of Wikipedia.

#👟Distance Traveled

On the other hand, distance traveled is a measure of the total path length covered by an object. It accounts for the entire journey, regardless of the direction taken. Unlike displacement, which focuses on the change in position, distance traveled is concerned with the total ground covered.

To illustrate this concept, consider this analogy of forgetting your homework in the morning. If you leave your house, realize you forgot your homework, and return to get it, your displacement is zero because your final position is the same as your initial position, since you are still at your house.

However, your distance traveled is the sum of the distances covered during both trips, capturing the t...