The concept of “instantaneous” anything is easy to teach (“Take a high-shutter-speed picture of your speedometer”) and very hard at the same time. Case in point: There’s a question on the calculus test that gives an equation for the distance of a particle at a given time. The students are asked to do the usual things: Find the velocity and acceleration at t = 3, find the times when the particle is at rest, etc. One part that has thrown a few students for a loop is: *What is the acceleration of the particle when it is at rest? *Several students have responded to the effect of:

0 m/s/s. An object cannot be accelerating when it is at rest.

That’s actually a pretty common physical misconception, and one that takes some non-calculus thinking to overcome. When one hears the phrase “at rest”, one tends to think of something sitting still, and remaining motionless for an extended period of time. Likewise acceleration is something we feel, the push/pull we encounter when the car we’re riding in hits the gas or the brakes suddenly. Giving the answer above is the result of putting oneself in the position of the particle; and since when** I** am sitting still or lying down I feel nothing, I therefore cannot be accelerating.

Of course this is not the case. A ball thrown up in the air undergoes an acceleration of 9.8 m/s per second downward at all points in its trip, regardless of where it is or how fast it’s moving, even at the top of its flight path when it is momentarily stopped. Acceleration — the change in velocity — is what makes the ball eventually come *down*. But since we don’t usually think of “at rest” as a concept that applies to instantaneous speeds — just speeds sustained over time — resting means not moving means not accelerating. Maybe this is a case where physical intuition is being carried *too* far.

Technorati Tags: Calculus, Critical thinking, Math, Teaching

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Our physical experiences of acceleration in the horizontal versus the vertical are quite different, qualitatively. Jumping up and down on a trampoline or falling from a roof feels way different than whipping around a corner in a car. But these sorts of questions (balls tossed in the air, people on carnival rides in circular motion) don’t distinguish between horizontal and vertical. The analysis is simply based on differentiating the position function. So I agree that physical intuition isn’t all that useful in thinking about acceleration. But your students aren’t being tricked by differences in sensations; they failed to really think through the question, maybe because they lack the tools to do so. I believe that it’s actually spatial reasoning (and the ability to translate the physical situation or function into a rough temporal/spatial graph) that’s needed.

So I think that some

calculusthinking — not non-calculus thinking! — will easily answer the question, and help overcome the misconceptions: A ball thrown up into the air has a position graph that’s a parabola, a velocity graph that’s a line with -9.8 slope, and an acceleration graph that’s a horizontal line intercepting the axis at -9.8. The relationship between the function (position), its derivative (velocity), and second derivative (acceleration) are just as calculus tells us they should be. No intuition required.It may, however, help if we use the phrase “instantaneously at rest” or talk about the particle “momentarily coming to rest”. Then students can, perhaps, re-engage their intuition and remember a moment at the maximum height of a trampoline jump! But I think the heart of the question is what we really mean by “intuition” and how we can better teach the connection between functions, graphs, and their interpretations. That is definitely tough to do.