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enter image description here

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

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    $\begingroup$ Related No matter how short a period it would take time to apply the brakes, stop pushing etc. $\endgroup$ – Farcher Feb 2 at 6:48
  • $\begingroup$ Thanks a lot, that thread helped me a lot. I did some searching prior to posting it but didn't find that thread. If I had, I wouldn't have posted mine $\endgroup$ – π times e Feb 2 at 9:05
  • $\begingroup$ Even if it isn't instant, it can be very sudden, as when a bullet hits a wall. Or what-if.xkcd.com/1 $\endgroup$ – mmesser314 Feb 2 at 16:06
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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

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  • $\begingroup$ Thanks for the explanation. Yeah it makes sense that the change should take place over a period, no matter how small $\endgroup$ – π times e Feb 2 at 9:06
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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.

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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.

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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.

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  • $\begingroup$ I understood the second part of your answer, that when we start pushing a body suddenly with a large force, the body accelerates quickly over a very short time interval (0.1 sec or smaller). I understood this part. But I didn’t understand the first part of your answer. Yes if we stop pushing the body, friction will slowly bring its speed down to zero. But here I was talking of acceleration suddenly dropping to zero, not velocity $\endgroup$ – π times e Feb 2 at 10:11
  • $\begingroup$ Yes.. That's the point..when we stop pushing, friction is a force which keeps on acting on the body, and that causes a negative acceleration. But there's a catch. It does not start acting instantaneously. When you keep on pushing, accn is +, when you withdraw your hand, the accn drops (almost instantaneously) and then the negative accn of friction keeps on acting. So there is no scope of zero accn. And also the drop of accn when you withdraw your hand takes place in a finite time interval. $\endgroup$ – MKC Feb 2 at 11:28

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