Introduction

Welcome again fellow readers! Today, FiveHive presents the second subtopic in Unit 4 in AP Calculus. In this article, we will focus our discussion of real-world applications and contexts of derivatives to the specific case of straight-line motion.

Rectilinear Motion

The official AP Calculus Course and Exam Description for this subtopic states “the derivative can be used to solve rectilinear motion problems.” Before diving into the actual calculus part of this, I want to take a moment to define linear motion. Linear motion involves the motion of particles moving on a straight, one-dimensional path (essentially a straight line). All AP problems relating to particle motion have this as the premise, so they won’t throw any curveballs at you. It is just helpful to know what assumptions you are working with here. 

AP exam questions often love to define the premise of these particle motion particles as particles being placed on either the $x$-axis or $y$-axis. Since those are straight lines, the motion would thus be linear. Motion along the $x$-axis would be horizontal, meaning the particle moves left/right. Motion along the $y$-axis would be vertical, meaning the particle moves up/down. The exam expects you to be able to naturally figure this out, and will not explicitly state this. It is important to understand this idea when interpreting and applying your calculus knowledge come exam time.

The Actual Connections

Ok, with that out of the way, we will now connect the aforementioned ideas of position, velocity, and acceleration. These are all characteristics that can quantify and describe different types of motion. Since motion changes over time, this means that position, velocity, and acceleration can all be expressed as mathematical functions of time. 

Position describes the precise location a particle exists in at any moment in time. It is often expressed as $x(t)$ and/or $s(t)$. Position can either be negative or positive.

Velocity describes the rate change of position with respect to time. Based on the work we did in AP Calculus 4.1, we can conclude that velocity is the derivative of position with respect to time. Velocity is often expressed as $v(t)$, and $v(t)=x’(t)$. It should also be noted that velocity accounts for both magnitude and direction. Unless otherwise stated, negative velocity means the particle is going left/down and positive velocity means the particle is going right/up. 

Acceleration describes the rate change of velocity with respect to time. Again, based on the understanding of AP Calculus 4.1, acceleration is the derivative of velocity with respect to time and also the second-order derivative of position with respect to time. Acceleration, often expressed as $a(t)$, is subsequently $a(t)=v’(t)=x’’(t)$. 

If you are confused on the units of position vs time, velocity vs time, and acceleration vs time, think back to Topic 4.1. The position vs time function, let’s use $x(t)$, outputs something with a position unit (for this example, let’s say meters, $m$) based on the time (for this example, let’s say seconds, $s$). If we take the derivative of the position vs time function to find the velocity vs time function, denoted by $v(t)$, the units would be $\frac{\text{units of }x(t)}{\text{units of }t}=\frac{m}{s}$. This is the unit of our velocity function. If we take the derivative of the velocity vs time function to find the acceleration vs time function, denoted by $a(t)$, the units would be $\frac{\text{units of }v(t)}{\text{units of }t}=\frac{\frac{m}{s}}{s}=\frac{m}{s^2}$. If you ever forget how the three functions are related, look at the units! Ask yourself it they make sense.

Other Important Motion Information

AP questions often want people to determine the time $t$ in which the particle changes direction. Based on the above logic of velocity sign being an indication of direction, this means that the instant in which a particle changes direction is when $v(t)$ changes signs. It is very important to note that if $v(t)$ does not change signs, then the particle does not change directions. Make sure to always write the statement “$v(t)$ changes signs” as a part of your justification if asked this in an FRQ.

Very Important: When AP Calculus problems ask if a particle is speeding up or slowing down, you need to check the signs of $a(t)$ and $v(t)$. If the signs of both are the same, then the particle is speeding up. If the signs of both are different, then the particle is slowing down. This is also the justification you will use in FRQs.

Position vs Distance & Velocity vs Speed

The distinction between the four quantities does matter. Going back to the fundamental definition, position describes precise location and can be positive or negative. Distance describes how many position units apart two locations are from each other. The crucial difference between position and distance is that distance can never be negative.

The similar idea applies for velocity and speed. Speed describes how fast something is going and does not account for direction. Essentially, it is the derivative of distance with respect to time and can never be negative. Thus, $\text{Speed}=|v(t)|$. 

Practice