This is part 1 in a 3 part series about how I’ve decided to move teaching projectile motion from one of the earliest units I teach, until after my students have a thorough grounding in vectors, Newton’s laws and momentum.

Projectile motion is the meat grinder of physics—more than any other topic, when adults or students recount the horrors of their experience with physics (this is a common conversation for most physics teachers) it centers around figuring out where an object hurled into the air at 30 m/s at an angle of 20° will land. If you examine the timeline of a traditional textbook, it’s easy to see why. Here’s how most of them go:

1. Units and lots of boring crap. Yes, there’s a some platinum bar stored under vacuum sitting in France that we use as a a kilogram. The real reason for this is never fully explained, nor is the awesome story of all the efforts we’ve gone through to rid of that dumb bar. Many teachers just skip this whole unit.

The kilogram

2. Kinematic equations. Yep, so you’re not really comfortable with albegra, and don’t really understand rates of change. Well, get ready cause it’s time to know the difference between a rate of change (velocity) and a rate of a rate of change (acceleration). And don’t forget, in physics world, acceleration can mean slowing down just cause we say so, and soon we’ll show you how you can even accelerate with constant speed. But for now you need to memorize these four (or five equations if you want to see how bad it gets over at sparknotes) equations:$\begin{array}{rcl}x&=& x_0+v_0t+\frac{1}{2}at^2\\ v&=&v_0+at\\ v^2&=&v^2_0+2a(x-x_0)\\ x&=&x_0+\frac{1}{2}(v_0+v)t\\ \end{array}$

Sure, the professor/teacher tries to show how these equations are all connected, and can be seen graphically, but students usually don’t get enough time to understand these ideas, and most of this stuff gets lost in a wave of tedious problem solving. Did I mention that many courses cover these topics in a week or less.

It’s quite possible that

$x=x_0+v_{0x}t+\frac{1}{2}at^2$

is the most complicated equation students will see in first year physics. For the untrained eye, this looks like an equation with 5 variables, superscripts and subscripts?! It is truly a mathematical monster.

3. Vectors. I hope you got those equations from the last chapter down, because now it’s time to learn that all those things you learned about in the last chapter have direction, and 2+2 no longer equals 4. Again, this can sometimes be covered in less than a week.Winded yet? In a college level class, you may have only been in class for 3 or 4 days and your head is swimming with all the deltas, vector symbols and equations. Learn fast, because it’s time for projectile motion.
4. Projectile motion. Here’s where the wheels fall off the physics wagon for many students. You start by telling students everything falls at the same rate, a dropped bullet and a fired bullet will hit the ground at the same time, and that if you’re hunting monkeys in trees, it is best to aim at them. Why are these things true? Who knows? The equations are the important thing, so focus on memorizing those. Then it’s time for a lot of hard problem solving, which can get a bit easier if you always remember to use the constant velocity equations for the horizontal motion, and the constant acceleration equations for vertical motion. You did figure out the difference between acceleration and velocity in the past two days, didn’t you? I hope you really like systems of equations and are pretty good at sorting out these equations that seem to have so many variables. And here’s the real kicker, almost nothing you do in these calculations is true. The baseball hit at 30 m/s and 15° will never land where your calculation says it will. Subtlety, you learn that the world of physics and the world of reality have little in common. But don’t worry, you’ll only be doing projectile motion for another few days before many courses jump on to Newton’s laws.

## The missing piece

The big problem with this treatment is it misses the beautiful picture of physics that many different phenomena can be explained with a small number of ideas. Falling bricks, hunted monkeys, batted baseballs, and even orbiting satellites can all be described with the very same principles. Why do the all objects fall with the same rate? Why should you shoot at the monkey? Why does the fired bullet hit the ground the same time as the dropped one? It’s because in all three cases, the objects are only experiencing the gravitational force, and this force is proportional to the mass of the object, so all projectiles will have the exact same acceleration. The only difference will be the parameters, starting position and velocity.

This is a subtle and beautiful idea that students can’t possibly get when they haven’t been exposed to Newton’s laws, and it is something that is hard to pick up if you’re flying through the standard physics text faster than most of the projectiles in the textbook problems.

While I’ve intuitively seen the pitfalls of the traditional approach to projectile motion for a while, I’ve found it very hard to break students out of the habit of approaching it from a traditional viewpoint. My students always want to default to equations to memorize, and this prevented them from having any real synthesis of the big picture, no matter how long I would wait to introduce projectile motion, and no matter how much I would emphasize starting with Newton’s laws to study it.

it wasn’t until this year, when I fully embraced modeling, that I came up with an approach to projectile motion that really seems to make sense, and gives students a strong feeling of the power of physics, and their ability to use it to understand the world. Modeling is a huge part of this, but it’s also a great opportunity to connect with computational thinking.

This will be the topic of parts 2 (Introducing projectile motion with Angry Birds) and 3 (Modeling real world motion with vpython) of this series.

1. February 16, 2011 7:34 am

I agree with lots of this, for sure, and have sequenced in some different ways in the past. I started one year (or two, I don’t remember) with constant v motion, which isn’t so bad, and then did N’s laws for objects with constant v motion. Then, a month later, we were back to kinematics, but now with acceleration, and then we looked at why the objects accelerate with N’s laws again. It was OK.

My big thing here about going on an equation-less romp through exclusively graphical and modeling analysis of kinematics – as conceptually superior as it would probably be, even though we cover those conceptual things now (which many don’t appreciate) – is what they do with kinematics later in the course. I want them to be able to see kinematics as a tiny, easy to crank part of a larger problem, since it usually is just that. When they hit a projectile motion at the end of a collision or some such, I’m usually interested in the collision, and like that they can crank out the proj. motion without much emotional investment (not without much understanding). Maybe that doesn’t make sense, but it’s a fear.

My second issue is about algebra: the biggest hurdle that I get kids over in the beginning of the course is doing algebra without numbers. Modeling x, v, and a functions and switching between graphs and equations are hard enough for the kids – can they do it with no numbers? Maybe so, and maybe it’d be a good hammer on modeling data with equations, too. Hmm… hard to say.

Thoughts?

• February 16, 2011 8:35 am

John,

This is funny, because sadly it’s true. Awesome piece. Should submit to The Physics Teacher. Seriously.

• February 16, 2011 8:54 am

jg,

The constant velocity –> NLs for constant v –> constant acceleration –> NLs for constant acceleration is the same sequence John and I are doing now. My students are solving goal-less kinematics problems without any formal kinematic equations, except for v_avg = disp/time and acc = delta v/time. (Which, even then, I don’t really introduce them as formal equations. We reason proportionally.)

For a goal-less problems, students must model the scenario completely and as quantitatively as possible: motion diagram, x-t graph, v-t graph. They find that they bounce back and forth between representations as they fill in the pieces. For example, it’s easy to label the motion diagram with the velocity at every second. But to get the position of each dot, they move to the v-t graph and calculate the area for each second. Almost all problems can be solved with v-t graphs, though some require simultaneous equations. And while you might argue that graphical problem solving is STILL plug-and-chug, it is at least not as intimidating as the kinematic equations.

To your point about kinematics being a smaller easy-to-crank part, I think it is still easy if the kids use a graphical approach. And about algebra without numbers — it think that is an abstract thinking hurdle few “regular” kids can successfully clear. Have them do it with numbers — I think having numbers with units allows kids to reason their way through a problem rather than symbol manipulation.

• February 16, 2011 8:32 pm

Frank, I totally agree. The one thing I might add is that sometimes, I think symbolic manipulation, instead of plugging in numbers can be a bit easier, since kids don’t have to track units all the way through their work.

• February 17, 2011 7:34 am

Well, they don’t have to write them through the work, but they’d better be checking them – at least before plugging in the numbers. It’s one of those things that we tell them will save them, show them how it could’ve alerted them to bad algebra, and some kids buy into, but others just never will!

• February 17, 2011 7:32 am

Probably our first reader – thanks! 🙂

I do like the goal-less problem idea, and I’m going to incorporate it next year. We start (after working with constant v for a few days) with modeling motion detector graphs of accelerated motion, and gradually develop the kin. eq. It’s more effective than deriving them (or having them derive them) from the def’ns, but I need to spend two days in the modeling process next year, so that I don’t have to push them along much with the interpretation of the parameters.

I follow much more of the conceptual/graphical/limited eq. approach with the regular physics students, and we don’t do kinematics until the end of the year, but I am beyond committed to exclusively symbolic algebra for the honors students. They can do it (after the two weeks of initial resistance) and don’t give it a second thought after the first month. It’s just so much more powerful.

Wherever we start, there’s lots that’s left out and stuff that we wish that they knew already. That’s part of the fun!

• February 16, 2011 8:30 pm

I followed pretty much the same order:

1. constant velocity
2. balanced forces (N1 & N3)
3. constant acceleration
4. unbalanced forces (N2)
5. conservation of momentum
6. projectile motion

I think this order has made a huge difference in my students’ understanding. Spacing out velocity and acceleration by a unit, and introducing the idea of force really gave my kids many more ways to approach problems and a much greater grasp of the difference between acceleration and velocity.

Our approach isn’t at all equation-less. Instead, I’m focusing on getting my students to see the meaning of the symbols they’re manipulating in equations. In an earlier post I wrote about how we worked to derive the constant kinematic equation directly from a velocity vs time graph. I keep coming back questions like what does $v_{0x}t$ and $\frac{1}{2}at^2$ mean? How are these quantities represented on the graph.

I’m working very hard with my honors students to build some better fluency with symbolic manipulation, and not plugging in numbers until the very last step. I’ve got a post in the works on how we used this to make some good sense of the monkey hunter today.

balanced forces (N1 & N3), constant acceleration,

2. February 16, 2011 8:55 am

What I’ve found most interesting about discussions like this is the sequencing of topics in textbooks. We seem to be forever bound to kinematics first, dynamics second. We see velocity and acceleration before we see vectors. We see kinematic equations before we have even read about forces. All the major texts (conceptual, algebra-based, and calculus based) are guilty of this. Talk with students about why motion can occur through pushes and pulls just seems to make more sense.

Newton’s Laws are actually quite general and extremely powerful. Why not convey this first and leave the killing of poor little monkeys in trees for later?

• February 16, 2011 8:34 pm

Danny,
Absolutely. This is the big lesson I’ve taken from M&I and tried to put to work in my modeling curriculum. Rather than stressing dozens of equations, we focus on a small number of models, and kids are able to see the power of a small number of ideas to explain a wide range of phenomena.

February 17, 2011 1:57 pm

I am a student in Dr. Strange’s EDM 310 class at the University of South Alabama, and i was assigned to comment on this post. I agree with you on this post, because I unfortunately took Physics in high school. Even though I did fairly well I felt like I learned so little. This is because I was so caught up in equations and problems that I never took the time to let it soak in. I believe by reading that you have figured out a productive way to teach Physics to make the students understand more, and I look forward to reading more. You can visit my blog at http://laforcebrentonedm310.blogspot.com/. Thanks.

4. April 28, 2011 1:22 pm

Just got around to reading this. I really like it. There is a professor at here at UMaine who teaches all of forces first (statics and then dynamics) and then goes to all of kinematics. I think your structure is probably better, and the reason, for me, is because it focuses on “let’s work on describing something” now “let’s work on explaining it”… and then again “let’s work on describing something a bit different” and then again, “let’s work on explaining this by extending what we already know.” and so, on… by the time you get to projectile motion, you’ve focused a lot of attention on explaining and not just describing. I think this fits in with my notion of science learning as helping students build more and more globally coherent explanations…

• May 2, 2011 9:19 pm

Brian,
I think this is exactly right. Just this year, thanks to Kelly O’Shea’s idea, I decided to teach kinematics by doing a unit on constant velocity, followed by a unit on Balanced Forces (N1 & N3), then constant acceleration, followed by a unit on unbalanced forces. This made a huge difference in understanding and gave them enough separation between velocity and acceleration to truly understand the two.