This has puzzled me for a long time.

enter image description here

Consider an arbitrary object's motion in this graph. It has to accelerate each time its slope changes, but there is no curvature in the graph, which is a dead giveaway for acceleration in pos/time graphs. Its speed is constant here always, with the exception of at $t=0$, $t=2$, and $t=3.5$ (roughly). Its change in speed also seems nearly instant. I know this is an idealized object, and real motion would never be able to do this (no object can go from $5 \ m/s$ to $-15/3.5 \ m/s$ (roughly) instantaneously), but this kind of motion graph always puzzled me. It's taught pretty early on in kinematics yet it is glaringly unrealistic and unintuitive, I find, despite its simplicity.

  • $\begingroup$ What do you think actually happens at 2s? $\endgroup$ – tfb Aug 4 '17 at 7:07
  • $\begingroup$ @tfb I think its velocity changes from that of a positive one to that of a negative one instantly, in, for lack of a better word, time $dt$. It had a large change in velocity in a hugely tiny amount of time. $\endgroup$ – sangstar Aug 4 '17 at 7:11
  • $\begingroup$ then I think you need to work on your physical intuition. This is not meant to be as rude as it sounds: intuition is important, and since you've asked this question you obviously have it. What it should be telling you is that this can't happen: this picture describes something not physically possible. In fact, around 2s, if you looked closely, there would be a finite but high acceleration -- perhaps the object has been hit with a hammer or something. So around 2s the graph is smooth: the 1st derivative of position is always continuous: it must be a $C^1$ function of time. $\endgroup$ – tfb Aug 4 '17 at 8:04
  • $\begingroup$ @tfb No offensive taken, however my gut is telling me this system is impossible and cannot happen, as I said the object cannot make jumps in its velocity instantaneously. I was just wondering if my thinking there was wrong. $\endgroup$ – sangstar Aug 4 '17 at 8:25
  • $\begingroup$ No: you are completely correct. If I taught physics I would have strong objections to graphs like this, as I think they actively damage intuition: half or more of the problems people have with the twin paradox in relativity is down to exactly this sort of misleading picture. $\endgroup$ – tfb Aug 4 '17 at 8:28

You are quite correct that as the graph is drawn the velocity is discontinuous at $2$ and $3.5$ seconds so the acceleration $dv/dt$ is undefined there.

But as you say this sort of graph tends to be used in elementary courses where we don't usually go into that level of detail. If challenged I would simply say the acceleration is so large that you can't see the curved region of the graph at this scale. That is the velocity change is effectively instantaneous.

  • $\begingroup$ What do you mean by "instantaneous acceleration"? $\endgroup$ – ccorbella Aug 4 '17 at 7:55
  • 1
    $\begingroup$ @ccorbella: sorry my wording was a bit unclear. Does it make more sense now? $\endgroup$ – John Rennie Aug 4 '17 at 8:08
  • $\begingroup$ Yes, thanks. In fact you are saying the same, but now I get it. Sorry, it must be just me. xD $\endgroup$ – ccorbella Aug 4 '17 at 8:11

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