Calculus is about things that are changing. Certainly, things that move are changing, changing their position, velocity, and acceleration. Most calculus textbooks deal with things being dropped or thrown up into the air. This is called uniformly accelerated motion since the acceleration is due to gravity and is constant. While this is a good place to start, the problems are by their nature somewhat limited. Students often know all about uniformly accelerated motion from their physics class.
The Advanced Placement exams take motion problems to a new level. AB students often encounter particles moving along the x-axis or the y-axis (i.e. on a number line) according to a function that gives the particle’s position, velocity, or acceleration. BC students often encounter particles moving around the plane with their coordinates given by parametric equations or their velocity given by a vector. Other times the information is given as a graph or even in a table of the position or velocity. The “particle” may become a car, or a rocket or even chief readers riding bicycles.
While these situations may not be all that “real”, they provide excellent ways to ask both differentiation and integration questions. but be aware that they are not covered that much in some textbooks; supplementing the text may be necessary.
The main derivative ideas are that velocity is the first derivative of the position function, acceleration is the second derivative of the position function and the first derivative of the velocity. Speed is the absolute value of velocity. (There will be more about speed in the next post.) The same techniques used to find the features of a graph can be applied to motion problems to determine things about the moving particle.
So, the ideas are not new, but the vocabulary is. The table below gives the terms used with graph analysis and the corresponding terms used in motion problem.
Vocabulary: Working with motion equations (position, velocity, acceleration) you really do all the same things as with regular functions and their derivatives. Help students see that while the vocabulary is different, the concepts are the same.
Function Linear MotionValue of a function at x position at time tFirst derivative velocitySecond derivative accelerationIncreasing moving to the right or upDecreasing moving to the left or downAbsolute Maximum farthest rightAbsolute Minimum farthest leftyʹ = 0 “at rest”yʹ changes sign object changes directionIncreasing & cc up speed is increasingIncreasing & cc down speed is decreasingDecreasing & cc up speed is decreasingDecreasing & cc down speed is increasingSpeed absolute value of velocity
Here is a short quiz on this idea.
Revised and updated from a post originally published on November 16, 2012