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.