In other words can the acceleration change instantly? In direction and/or magnitude.

There are two aspects to this question.

  1. In a problem, can you treat acceleration as changing instantly? (when in reality the change is actually finite but large)

  2. Is there any way that acceleration can change instantly. Not as an approximation, but actually change instantly.

It seems the answer to 1) is yes. And the answer to 2) is no.

I can accelerate a block by pulling on a cord and at then just let go. The acceleration can be modeled as going to zero instantly. In this case, it seems that seems that the answer to 2) is also yes.

Velocity can never change instantly. That would mean that acceleration diverges and this is not physically possible. Position can't change instantly for obvious reasons.

This question has been explored here: https://www.reddit.com/r/askscience/comments/4eadfj/can_acceleration_change_instantaniously_or_is_it/


1 Answer 1


According to the least action principle, the behavior of the system depends on positions and velocities, but not on accelerations (per the Euler-Lagrange equations). Therefore, theoretically, an instant change of acceleration is hypothetically possible, as it does not violate any conservation laws. However, no real measurable value can be infinite. Therefore, in reality, a displacement derivative of any order cannot diverge. The limitations would appear technical, as in the object not being infinitely rigid or the system inability to apply a full force instantly, etc.

Thus the answers to your questions are yes to 1 and no to 2.

  • $\begingroup$ Also, I feel that the division between 1 and 2 is a bit inchoate. Classical mechanics is an approximation in the first place. (And there are degrees of approximation within classical mechanics). Newtons laws or Hamiltons principle puts no limits on any quantities. But no limits does not mean a quantity can be infinite; but it can be so large that it can be approximately (or relatively) infinite. Note the Lagrangian can contain higher order derivatives. Not sure that Hamiltons principle restricts this. $\endgroup$
    – yalis
    Jun 13, 2018 at 4:53
  • $\begingroup$ There also would be an increasing uncertainty in measuring higher order derivatives. I think the highest meaningful order is not very high (single digits). Jerk is a big issue in a public transportation (I hate electric buses for it), but higher orders quickly become unintuitive. $\endgroup$
    – safesphere
    Jun 13, 2018 at 8:30

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