# Acceleration at maximum displacement

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1. Sep 20, 2016

### kuruman

A common student misconception is that when a ball is thrown straight up in the air, at the point of maximum height, where the velocity is zero, the acceleration is also zero. This can easily be dispelled by observing that, if indeed the acceleration were zero when the velocity is zero, the velocity would not change and the ball would remain at maximum height forever. We all know this does not happen and that under the constant acceleration of gravity, what goes up must come down.

All this is good, but I wonder to what extent the issue is befuddled when students encounter the example of a block that, when given an initial kick, slides on a horizontal table until it comes to rest. In this case we teach that the net force is kinetic friction and equal to μkmg. Therefore, the acceleration that opposes the velocity is also constant and equal to μkg. At this point the proverbial astute student might ask, why does the ball in the previous example return back to where it came from whereas the block in this example does not? After all, they are both moving under a constant acceleration that initially opposes the velocity, are they not?

Yes, but ... the expert understands that the constancy of frictional acceleration is only an approximation and that the force known as friction is phenomenologically velocity-dependent and goes to zero as the velocity goes to zero. In other words, the acceleration of the block is constant until it is not. Is this confusing to the novice or what?

When I teach kinetic friction, I address this apparent paradox by juxtaposing the ball and block examples and explaining that the constancy of the block's acceleration is only an approximation, valid for a range of velocities until the block is just about ready to come to a stop. At low speeds, there is "stick and slip" behavior with progressively more "sticking" until there is no more "slipping."

It's not bad practice to pretend that the acceleration due to friction is constant. It is good practice, however, to clarify that an object achieving zero velocity under the influence of a net conservative force has reached a turning point while an object achieving zero velocity under the influence of a net non-conservative force comes to rest and remains at rest. Perhaps this is a more intuitive way to illustrate the difference between conservative and non-conservative forces to students whose exposure to the definition of work is mostly recent.

2. Sep 20, 2016

### robphy

To help with such topics that are confusing to a novice, it helps to give a general framework for setting up and analyzing the situation.
So, a free-body diagram [actually, a sequence of free-body diagrams] is needed.
In addition, that nature of the forces are important...
the gravitational force vector is constant (near the earth),
whereas the friction force vector changes magnitude and direction.

Possibly useful:
https://phet.colorado.edu/en/simulation/legacy/forces-1d
https://phet.colorado.edu/en/simulation/legacy/forces-and-motion
https://phet.colorado.edu/en/simulation/the-ramp
https://phet.colorado.edu/en/simulation/legacy/ramp-forces-and-motion

3. Sep 21, 2016

### Andy Resnick

Lots of good stuff, I like your approach. Another approach (for intro- style courses) is to broadly classify forces as 'active' or 'passive'. 'Passive' forces, such as the normal force, only exist in response to active forces and adjust themselves accordingly. Using your example, if the block is initially on a level surface and at rest, there is a normal force and no friction force. If you then incline the surface (not enough to allow the block to slip), the direction of gravitational force changes, and in response both the normal and frictional forces change such that the net force is still zero.

4. Sep 21, 2016

### kuruman

Absolutely. Instead of "passive" I call them "contact" forces and I qualify "accordingly". My statement is that "contact forces adjust themselves to provide the observed acceleration, but only up to a certain point." Blocks start sliding, tables collapse and strings break if we demand that they provide a contact force that exceeds its upper limit. When I ask my students to give me the direction of the force of static friction that accelerates them when they start walking from rest, about 3/4 of them say it is in the opposite direction of the motion. An exercise as simple as walking is a prime example of a contact force that adjusts itself to provide the observed acceleration.

5. Oct 12, 2016

### Kajal Sengupta

I have a different approach to explain this. Gravitational force is not velocity dependent and would act even if the body is at rest ( for example a box on the ground). So when the ball reaches the maximum height and its velocity is zero, the gravitational force will continue to act in the same direction and hence the ball comes back to ground. In case of box example in the original question kinetic friction ( whose origin is electromagnetic forces between the surfaces ) will be active only if the block is in motion and ceases when it comes to rest. Thus once the block is at rest it can not return to original position because the force ceases to act.. Please correct me if I am wrong.

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