AP Physics C 2002 M2 Question

Hi All -

2nd year in my teaching career and already I'm teaching an AP C: Mechanics course! As I'm creating my midterm for my students, I'm looking at AP Free Response questions to give them practice for the real thing. As we're only 8 units in, my selections are limited to Work, Energy, Dynamics, and Kinematics essentially. That said, I've found what I find to be a great resistive motion problem.

The exam is on AP's website http://apcentral.collegeboard.com/apc/members/repository/physics_c_m_00.pdf".

My problem is concerning the last part of Question 2 in the Mechanics section. It reads "Determine the energy dissipated by the drag force during the fall if the ball is released at height h and reaches its terminal speed before hitting the ground, in terms of the given quantities and fundamental constants"

I can easily see how my students would produce the answer of "Energy dissipated is what would be predicted by conservation of energy - what is actually there" or "mgh - 1/2mv2. My problem is with the second way to find the solution as Work = integral F dot dr. If the force is given by bv2 then we are integrating bv2. However, the velocity should change with respect to height, so this is no easy integral. I'm struggling to see how to approach the integral without finding a function v(x).

Can anyone help?

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K^2
Energy conservation. You know terminal velocity, so you know final kinetic energy. The rest of the potential energy goes into work against drag. You don't have to integrate anything.

Energy conservation. You know terminal velocity, so you know final kinetic energy. The rest of the potential energy goes into work against drag. You don't have to integrate anything.

Right, but that's not the point. The AP board found it significant to accept an integral for work. I already mentioned that my students can do it that way, but I want to make sure that when I'm explaining the ways to do it, I provide them with both ways in case someone doesn't think about conservation. The rest of the problem was about forces, so it's natural that their brains might be tuned to Work and forces here.

K^2
You can integrate over quadratic drag, but you first have to solve a differential equation for a free fall with drag, and that's not a trivial one. Once you know the answer, it's easy, of course, but it's not the AP-level stuff.

Given initial time t0=0, initial velocity v0=0, and terminal velocity vt, you can find instantaneous velocity at some time t.

$$v(t) = - v_t tanh\left(\frac{g t}{v_t}\right)$$

It's easy to convince yourself that at t=0, a=-g, and as t->∞, v->vt, and if you solve for vt in terms of b, g and m, you'll see that this v(t) satisfies m(dv/dt) = bv² - mg.

Power dissipation is bv³, so total work done by drag is integral of that over time from t=0 to impact. The integral of tanh³(x) is sech²(x)/2 + log(cosh(x)), which is not the sort of thing you expect an AP student to be able to compute in the first place. (Most of them aren't even familiar with basic hyperbolic functions, let alone methods of integration over these.) And, of course, then you have to find time to impact. For that, you have to integrate over velocity to find displacement over time. Integral of tanh(x) is log(cosh(x)). This one is a bit easier, but solving for the impact time is purely numerical work. There is no analytical solution.

So yeah, actually integrating over a drag force is nothing that a descent graduate student in physics should have difficulties with, but if you expect your AP students to follow all this, you are dreaming. If you think you got all that, you can try explaining it to some particularly bright and eager ones one-on-one, but if you try this with a class, you'll loose their attention, and there is nothing worse than that.

Go with the conservation of energy. Mention that you can solve this by actually integrating over forces using crazy hyperbolic trig, and leave it at that.