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## 26 November 2016

### Algebraic kinematics in CONCEPTUAL physics

Our discussion of motion in conceptual physics starts with a three-week unit.  The first week deals exclusively with position-time graphs.  The second week deals with velocity-time graphs.

The third introduces the definition of acceleration, along with three equations of motion.

Acceleration is defined as how much an object's speed changes every second, and is measured in units of m/s per second.  The direction of acceleration is the same as the direction of movement when an object speeds up; the direction of acceleration is opposite the direction of movement when an object slows down.  We never use the words "positive" or "negative" to denote directions.  Instead, we use words like left, right, up, down, forward, backward.  In the absence of a clearly defined direction, students are directed (hah!) to talk about "toward the motion detector" and "away from the motion detector."

Now this is conceptual physics, so the descriptive analysis above forms the basis for the students' understanding.   Nevertheless, there are considerable quantitative and semi-quantitative elements to this course.  We do get numerical answers for distances, speeds, and accelerations; if nothing else, recognizing the reasonability of a numerical answer is a major physics skill.

I do not present an "equation for acceleration."  Most texts state my definition of acceleration as an equation: a = Δv /t.  I don't like that.  It's too easy to miss the delta, and to say that acceleration is speed divided by time.

Instead, I ask students to reason in their minds using the definition of acceleration.  "Since it took 5 s for the cart to slow down by 20 m/s, the cart slowed by 4 m/s every second.  Its acceleration is 4 m/s per second."  This works both pedagogically -- in every problem I'm reinforcing the operational definition of acceleration -- and physically, because we are always assuming constant acceleration.

However, I *do* present equations for distance traveled.  Three of them, as a matter of fact.

In conceptual physics, we restrict our linear kinematics problems to three situations:

(1) Objects moving at a steady speed
(2) Objects speeding up from rest
(3) Objects slowing down to rest

In case 1, the distance traveled is given by d = vt.
In cases 2 and 3, the distance traveled is given by EITHER d = (1/2)at2 OR d = v2/2a.

In previous units, we've taught our students about semi-quantitative reasoning (i.e. when resistance doubles at constant voltage, what happens to current?) and about predicting the shape of graphs (i.e. sketch a graph of resistance vs. current at constant voltage).  We also have been rigorous about beginning any direct calculation with a chart of values with units.  Here we have three new equations on which to practice these skills.

Plus, we learn a new skill: picking the applicable equation out of a lineup.

"But that's not a skill, Greg, that's easy," you say.  Nuh-uh.  Students raised through the mathematical-industrial complex are rarely taught to consider the situations in which an equation might be valid or invalid; and though they're taught how to match variables with relevant values (in "word problems"), most have in no way mastered that skill.

Again and again conceptual physics students carefully calculate the distance an object traveled... by picking one of the three equations at random, while randomly assigning values from the problem statement to the variables in that equation.  Then the students are upset that I don't accept their answer.  They did the math right!

So we work hard on identifying the right equation to use.  "Is it constant speed?  Then you must use d = vt.  No, you can't use d = (1/2)at2 , it doesn't matter that we know the time and the distance, you can't solve for acceleration when something moves at constant speed, and the problem asked for the speed, anyway."  "It's slowing down to rest and we know the acceleration.  Both d = (1/2)at2 and  d = v2/2a are valid.  Do we know the time it took or the initial speed?  Because then we'll know which of the equations is useful."

After this unit the students are only kinda okay at doing semi-quantitative and quantitative kinematics.  That's fine.  Because pretty soon we'll revisit these equations in new contexts.  Perhaps we've used Newton's second law to solve for an object's acceleration, and we want to know how far it goes in 4 seconds.  Even better, once we understand how to deal with velocity as a vector, we study projectile motion for objects shot horizontally off a cliff -- where d = vt is valid horizontally, and the other two equations are valid vertically.