Instantaneous changes in acceleration

Question

These are very simple looking graphs. What I don’t understand is (I’m talking of the highlighted text in blue at the bottom), it says such instantaneous changes in acceleration can not occur in reality. If acceleration is positive for a certain time interval, can’t it be zero instantaneously if we stop pushing the body, i.e if we stop applying force on it ‘suddenly’? Similarly, if a body is moving with a constant velocity (acceleration is zero), and if we apply brakes suddenly, doesn’t its acceleration become negative instantaneously? I can’t seem to understand why the third graph is not possible, as the book says.

And, if the third graph is not possible, it means the two graphs above it are not possible either, are they? Because all three graphs here describe the same physical situation. Please help me understand this. Thanks

Answer 1

Sorry for my poor english !

I share your interrogations about this remark.

In physics, we are always working on modeling. What is the mathematical object I use to describe the system? By Newton's law, a discontinuous acceleration is a discontinuous force. Should we accept discontinuous forces in our modeling?

A discontinuous force, $F(t)$, is a force that I see vary from a finite quantity in a zero time. In physics, we always have a temporal resolution of the measuring system. This force varies very quickly, on a time less than the resolution in time of my measurements.

One could say, "if I improve the resolution in time, I will see that the function $F(t)$ is actually continuous".

For the car that brakes it is surely true. But in general, it's not sure ! One could imagine a time scale so short that one has to bring in quantum mechanics, or the Brownian movement .... and have to abandon the notion of force before seeing it as a continuous function.

One can imagine the same situation in electrostatic conductors. Surface densities of charge are used. One could say, "in a finer modeling, these densities are volume densities". But it may happen that by decreasing the spatial scale, the atomic structure appear before having been able to consider the surface charges as distributed in volume.

Answer 2

We define acceleration as the rate of change of velocity and velocity as rate of change of position. For v=dx/dt to be valid x has to be a continuous function of t. Similarly for a = dv/dt. So velocity changing abruptly as in the picture would be a problem.

Practically also any real change in velocity or position will take a finite time.

In view of these problems, when such idealized situations are described to illustrate physics ideas, the limitations are also mentioned. The highlighted text is one such example

Answer 3

This happens in quantum mechanics when an electron in an atom is excited to another state. This happens instantaneously. The acceleration towards the centre is changing abruptly since the possible states of the electron is quantised. So at-least in QM, acceleration can change abruptly.

Answer 4

I disagree. Acceleration can change instantaneously.

Imagine a car that is decelerating with constant force applied to the brake. But cars are complex, what with shocks and bendy metal, etc. Focus on just the wheel. But even a wheel isn't exactly a spherical cow. Let's look only at the brake pad and the disc. In fact, we will only consider the relative velocity of the portion of the disc at the interface with the pad.

As the disc spins under the brake pad, constant sliding friction reduces the angular velocity by some constant amount. The disc is accelerating at some negative value, which depends only on the amount of force on the brake and the coefficient of sliding friction. As long as the disc is moving at all, this acceleration will not change.

However, when the angular velocity reaches zero, the coefficient of sliding friction is no longer relevant. The disc goes from being "in motion" to being "not in motion." It doesn't gradually transition from one state to the other. It doesn't coast down to smaller and smaller values. The friction instantaneously goes from sliding friction to static friction. Thus, the acceleration changes from some negative value to zero instantaneously.

Answer 5

Let us have a good look at the third graph. The acceleration suddenly becomes 0 at time Ta seconds.

Now let us return back to the real world. Imagine you are pushing a block with a force which causes an acceleration a in it. Now, when you stop pushing it, what happens? Does the acceleration drop to zero?

No, that is not the case. Friction keeps on acting on the sliding block, which which will slowly decelerate it it zero speed. When you start pushing a body suddenly, the body accelerates quickly, but not as shown. It takes some time, however small it may be(0.01 sec or even small)

So that type of graph is not possible.