Momentum bar charts: worked-out examples

A large circular disk is initially stationary on a horizontal icy surface.  A person stands on the edge of the disk.  Without slipping on the disk, the person throws a large stone horizontally at initial speed v_o relative to the ground from a height h above the ice in a radial direction, as shown in the figures above.  Consider the x-direction to be horizontal, and the y-direction to be vertical.  Consider the system consisting of the person, ball, and disk.  “Initial” refers to before the ball is thrown; “final” refers to the instant before the ball hits the ground.

The picture and some of the description is from an old AP Physics C exam question, which asked for detailed calculations of various quantities in terms of given variables and fundamental constants.  In AP Physics 1, there's no need to do the calculations;however, it's critical that we teach how to set up those calculations, or at least how to explain what is conserved and why.

I pose this and eleven other interesting situations in my energy and momentum bar chart exercises.  These are comprehensive, end-of-course activities that will challenge most physics teachers.  There's no algebra, none at all; just the requirement for careful understanding of the meaning of a "system", and then of an external force acting on that system.    

Can you make a qualitative impulse-momentum bar chart for the x-direction?

Of course you can.

Initially, nothing moves; so nothing has momentum.  No impulse acts on the person-ball-disk system.

What about the impulse due to the force of the person on the ball?

Since both the person and the ball are part of the system, the force of the person on the ball (and its Newton's 3rd law companion) are internal to the system.  The impulse column requires impulse applied by the net force external to the system -- only the net external force can change the momentum of a system.  In this case, there are no external forces in the horizontal direction.

The point of the bar chart is that it shows by inspection how the total system momentum is distributed: the bars on the left side plus the bars in the middle equal the bars on the left.  In this case, there must be zero total momentum after the throw.  How is that accomplished?  The ball moves right, so has what I'm calling positive momentum.  To maintain zero total momentum, the person and disk move left, giving them negative momentum.  The person and disk move together, giving them the same speed -- not the same momentum.  Since the disk is more massive than the person, the disk has a larger share of the system's negative momentum.  Note that the bars representing the person and disk add to about the same size as the bar representing the ball, showing that the total momentum remains zero.

How about the y-direction?

Sure, though it's a bit trickier.

Again, initially no movement or momentum.  Think for a moment: what causes the impulse in this case?

It's NOT the person pushing the ball.  That's a force (and thus an impulse) in the horizontal direction only.  And, that's internal to the system, anyway.

In the vertical direction, two external forces act on the system: the normal force of the ground on the system, and the force of the earth on the system.  The force of the earth is equal to the weight of the entire system; the normal force here is equal to just the weight of the person-disk part of the system (because the normal force is a contact force, and the ground is only in contact with the person-disk part of the system; the ball is in free fall).  So the net external force is equal to the weight of the ball.  That causes a downward impulse, represented by the bar in the chart above under the J.

Now inspect the bar chart: zero bars initially plus the impulse bar must equal the bars of total momentum when the ball is about to hit the ground.  The person and disk still don't move vertically, so they have zero momentum.  The ball must have a momentum equal to the impulse provided by the earth on it; that's represented in the chart by the ball's bar having the same size as the impulse bar.

That's enough for today.  But you can answer many, many more questions involving this situation.

What about an energy bar chart?  (Energy is a scalar, so you don't have x- and y- direction charts for energy.)  What if the earth is part of the system?  What if the system is JUST the person and disk?

See, the situation is rich, rich, rich with subtle questions.  Have fun with these.  Post thoughts in the comments.  Assign them to your students, and post the common misconceptions.  Go nuts...

GCJ