Introduction to vectors in general physics

Folks are often surprised that, in AP physics, my "vectors unit" consists of one problem, in which we show that a rope at an angle is equivalent to two ropes, one vertical and one horizontal.  We use sines and cosines to calculate the vertical and horizontal components of the rope's tension.  That's it.  And that's enough -- my class developes the skills necessary to break all kinds of vectors into components, and to add vector quantities, just as well now as when I used to do a week-long unit on vector math.

In general physics, I do a bit more mathematical prep work as we move into forces at angles.  But not much.

By now, my general physics students can deal with Newton's second law, as long all forces are either horizontal or vertical.  For example, we do problems with cars slamming on the brakes, and slowing due to friction.  Next, I want them to deal with forces at angles, like the lawnmower that's pushed along its handle. 

We start the process of breaking vectors into components with displacement vectors, not force vectors.  Why?  Because students are familiar with the concept of cardinal directions and shortcuts:  going 3 miles north and 4 miles west is the same thing as going 5 miles northwest, as long as you choose the angle correctly.  Everyone gets that.  If I can show them the mathematics in the context of displacement, then they can transfer those mathematics to force vectors.

Take a look at the assignment below.  Note that Will Collier is a real student, whom I taught in 2001-02.  He really does run a transport unit in Iraq, or at least he did last year.  The rest of the exercise is merely convenient science fiction.  I've done similar exercises before using maps of New York City, Berlin, and South Africa -- pick something of interest to one of your students!

Instructions for vector assignment:

Imagine that a new type of pilotless aircraft is developed. The plane is reliable and safe, yet inefficient; due to the vagaries of its design, this plane can fly ONLY along the cardinal directions: north, south, east, or west.

Will Collier, Woodberry class of 2002, runs a military transport unit. We will imagine that he has commandeered a large number of these aircraft to deliver goods throughout Iraq from a base in Baghdad. Your job as Lt. Collier’s unit navigator is to give directions to other Iraqi cities such that these special aircraft can fly there.

So, for each of the ten destinations you choose for Lt. Collier’s supplies, draw a displacement vector to the town.

Then, for each displacement vector, on unlined paper, using half a page per town, do the following:

• sketch the displacement vector and its components on x and y axes.

• state the magnitude and direction of the displacement vector.

• calculate the x and y components of the displacement vector, showing all work carefully as instructed in class

• write out, in words: "To fly to [whatever town], fly 200 km north, then 300 km west." Of course, fill in the correct distances and directions.

Follow these instructions (which are exactly what I said in class) carefully. I expect thorough, neat work. This should not take an exceptionally long time; work quickly but carefully. You should find this assignment to be the easiest all year. If you do not, come see me ASAP!

Conservation of momentum in the English Premier League

So, I watched the Sunderland-Liverpool match in the English Premier League on Saturday morning.  (I like to write my student comments in front of sports on TV.)  Just a few minutes into the game, Sunderland scored a strange goal.  Watch -- there's a real physics purpose here...

[Edit:  Looks like the EPL removed this video from youtube.  I'm sure you can find it somewhere... it's fantastic.]

[To summarize:  A Sunderland striker executed a shot on goal from about 15 yards out.  The shot hit a red beach ball-type object and deflected at an angle; an easy save for the Liverpool goalkeeper turned into the deciding goal in the1-0 match.]

The commentators originally called the unusual red object on the pitch a "balloon."  I wrote some sports-related commentary on this event on my sports blog,  "Nachoman's Baseball."  But imagine the physics possibilities here...

The assignment, which I will use as an independent experiment in the spring in AP physics, is:  Determine the mass of the balloon / beach ball / whatever that caused the goal. 

GCJ

Clicker activity -- basics of Newton's Second Law

My wife has been gone for four days. She's hiking with the sophomores on their Outward Bound trip to North Carolina. That means, however, that I'm in charge of six-year-old Milo all by myself. I quite enjoy occasional solo time with the boy. But what to do on Saturday morning, when I had to teach a physics class some more about Newton's Second Law?

I prepared a clicker exercise. Milo loves using the clickers, and he loves being part of a class with juniors and seniors. In turn, the juniors and seniors are welcoming and friendly to Milo -- even moreso due to the "Milo Questions."

I had Milo write a set of multiple choice questions about himself. I promised to use these as part of the in-class activity. Thus, the overall set of questions for the day's clicker exercise consisted of two Newton's Second Law questions, followed by one Milo Question, followed by two more second law questions, etc. The class (including Milo!) was divided randomly into teams; the team that got Milo was excited, because they knew that they had the Milo questions in the bag. As always, each team could collaborate and submit a single answer to each question. They earned one point for a correct answer, and one bonus point for each group who did NOT get the correct answer.

The actual set of questions is below. Feel free to use them. They may sound really easy, but remember how difficult it is to remember and assimilate even the most basic facts about the second law. It takes an amazingly huge number of repetitions before we can break down the most common misconceptions like "motion requires a force" and "acceleration tells which direction something is moving."

(I gave a "fundamentals quiz" about some of these same ideas a few days later. I'll try to post that soon.)

GCJ

1. A bucket whose mass is 10 kg hangs by a rope in which there is 63 N of tension. What is the weight of the bucket?
(A) 100 N
(B) 10 N
(C) 10 kg
(D) 100 kg
(E) 63 N
(F) 63 kg
(G) 73 N
(H) 73 kg
(I) 163 N
(J) 37 N


2. A bucket whose mass is 10 kg hangs by a rope in which there is 63 N of tension. What is the net force on the bucket?
(A) 37 N
(B) 163 N
(C) 100 N
(D) 63 N
(E) The answer depends on which way the bucket is moving.


3. What color is Milo’s house?
(A) Green
(B) Blue
(C) Purple
(D) Yellow
(E) Grey
(F) Brown
(G) White


4. A bucket whose mass is 10 kg hangs by a rope in which there is 63 N of tension. What is the magnitude [i.e. the amount] of the bucket’s acceleration?
(A) 6.3 m/s2
(B) 0.63 m/s2
(C) 3.7 m/s2
(D) 0.37 m/s2
(E) 10 m/s2
(F) 1.0 m/s2

5. A bucket whose mass is 10 kg hangs by a rope in which there is 63 N of tension. What is the direction of the bucket’s acceleration?
(A) Up
(B) Down
(C) The direction of acceleration is unknown


6. What does Milo do after seated meal?
(A) Go out back
(B) Go home
(C) Come here
(D) Go to bed


7. A bucket whose mass is 10 kg hangs by a rope in which there is 63 N of tension. What is the direction of the bucket’s velocity?
(A) Up
(B) Down
(C) The direction of velocity is unknown


8. So how could it possible for the bucket to move upward, then?
(A) The bucket must be slowing down
(B) The bucket must be moving at constant speed
(C) The bucket must be speeding up
(D) The tension has to increase to more than 100 N


9. How many bunnies does Milo have?
(A) 0
(B) 7
(C) 2
(D) 1

Centripetal vs. Centrifugal Force: Golf Cart

A golf cart is moving in a straight line. I want the cart to move in a circle. Should I push or pull the cart TOWARD the center of the circle, or AWAY FROM the center of the circle?

Of course, this is the central (ha!) question of the circular motion unit. Students have preconceived notions of "centrifugal force," as well as mistaken ideas about force in the direction of motion. It's nice to begin the circular motion unit with this central question, followed by a demonstration that shows unambiguously and memorably that force toward the center of the circle is required.

Since I live on campus, about 0.5 miles from my classroom, I drive a golf cart to work. This morning I blocked off about 10 spaces in the little parking lot next to the science dungeon. I tied a sturdy rope to the corner of my cart. With the class watching, I drove the cart forward. A physically strong student pulled on the rope in a direction perpendicular to the cart's velocity. Sure enough, the cart's path arced slightly.

Next, I had THREE students tug on the rope. This time the cart's path described a "tighter" circle. We will use this qualitative observation on Monday, when we write and use the equation for centripetal acceleration.

And finally, I turned to a student who originally answered that we should pull the cart AWAY from the center of the circle. I asked him to do so, but he smiled and politely declined. Woo-hoo -- he gets it.

GCJ

A quick thought for a busy October

October at Woodberry Forest is "death month" -- two parents' weekends (one for upperclassmen, one for lowerclassmen), grades and comments due, and teaching the heart of a front-loaded physics program lead to exhaustion. I actually asked IN to duty last night so I would stay awake long enough to grade my first general physics test.

So while I don't even have time for postseason baseball in October -- who does, when the games go on for 4.5 hours -- I can at least post a quick exchange from last Saturday's general physics class.

I brought out the classroom response system, the "clickers," because it was 8:00 on a Saturday morning. I gave everyone a velocity-time graph, and asked questions about it: "During which time interval(s) is the car speeding up?" "During which time interval(s) is the car traveling south?" I divided everyone into teams of two, giving each group a clicker.

Elliott asked, as someone all-too-often does: "So, are you going to count this for a grade?"

My stock response, which this class heard for the first time: "Why? Are you going to take it more or less seriously depending on my answer?" Elliott wisely kept his mouth shut.

There's a moral to that story, but I'm too exhausted this morning to figure out what it is. Good luck surviving your own October.

GCJ