Connecting Position, Velocity, and Acceleration of Functions Using Integrals

#8.2 Connecting Position, Velocity, and Acceleration of Functions Using Integrals

In this guide, we will look at how to use integrals to find the position, velocity, and acceleration of a function. We'll break it down for you in a way that's not just about numbers and formulas, but about understanding the language of motion. 👟

#💭 What are Position, Velocity, and Acceleration?

To begin with, it is important to understand the definitions of position, velocity, and acceleration. Going back to unit 4, we did cover rectilinear motion and applied our knowledge of derivatives to these three concepts. Check out our 4.2 guide here for a review! If you’re curious and want to learn more, we have an entire guide dedicated to these concepts as part of our AP Physics 1 study material.

#🧍Position

In the calculus world, position refers to the location of an object in space at a given time. Mathematically, we denote position as a function of time, usually represented by $s(t)$.

Imagine you're in the driver's seat of your dream car, and you want to know where you are on the highway. Well, that's your position $s(t)$. It's like your car's GPS telling you exactly where you are at any given time. Let's say you start at mile marker zero—that's your starting position. We can represent an object’s position as a function of time using a graph, such as the graph below.

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The position vs. time graph for an object.

Image courtesy of BCcampus Pressbooks.

Let’s say the graph above represents the movements of a caterpillar. From time $t = 0$ to $t=0.5$, the caterpillar is moving in a positive direction, away from the starting point. We can conclude this because the slope of the first line segment is positive.

At $t=0.5$, the caterpillar is at position 0.5 m—in other words, the caterpillar is 0.5 meters away from its starting point. Then, from $t=0.5$ to $t=1$, the caterpillar does not change position, meaning it has stopped moving and is at rest. Whenever there is a horizontal line on a position vs. time graph, it means the object is at rest.

Finally, from $t=1$ to $t=2$, the caterpillar is moving in the negative direction, back towards the starting point. We can conclude this because the slope of the last line segment is negative.

It is important that you learn to read this graph, as well as the graphs that are pictured below, to make sure you have a good understanding of the relationship between position, velocity, and acceleration. 🤯

#🚗 Velocity

Velocity is the rate of change of position concerning time. In simpler terms, it tells us how fast an object is moving and in which direction. Mathematically, velocity, denoted by $v(t)$, is the derivative of the position function $s(t)$ with respect to time.

Going back to our example of your dream car, let’s say you step on the gas, and your car starts zoo...