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I Is Acceleration/Decceleration smooth?

  1. Jan 3, 2017 #1
    Firstly I must say if this is in a wrong sub-forum I apologise. This is my first post and I'm new to the website so please bare with me. Also I was unsure of the Prefix so I again apologise if that to, was incorrect.

    But my Question is that is the acceleration (or deceleration) of an object smooth?
    By this I mean can the speed of an object increase instantaneously (for example) from 2Mph to 4Mph?
    Or must it be smooth in it's acceleration by passing all speeds in-between 2Mph and 4Mph to get to the end 4Mph?

    By passing all speeds in-between i mean such as it goes (in terms of speed) through every number in-between the 2 and 4Mph.

    Sorry if this is not worded well as I am unsure of how to, but I hope you understand and reply.
  2. jcsd
  3. Jan 3, 2017 #2

    Vanadium 50

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    In principle, acceleration can be instantaneous - hit it with a hammer. In practice, it always happens over some time - e.g. the steel hammerhead deforms ever so slightly so the acceleration is continuous over a very small time scale.
  4. Jan 3, 2017 #3


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    To have speed jump instantaneously from 2 mph to 4 mph would require infinite acceleration. We never get infinite acceleration, so the answer is no, speed can not jump instantaneously.
  5. Jan 3, 2017 #4


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    Smoothness means infinitely often differentiable, i.e. there are no vertices in the (position,time)-diagram and all of it differentials.
    You're description by the use of instantaneous describes the continuity of this diagram, i.e. no gaps.
    Although there might be no gaps, which would mean a division by zero, there still can be vertices, e.g. at the start when moving from zero to a constant acceleration.
  6. Jan 3, 2017 #5


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    There are two aspects of 'smoothness' assuming that you could apply an instant change in the force on an object, the velocity would not change instantly; the transition would be smooth. Also, an instantaneous change of velocity would not result in an instantaneous change in position - that would also involve a smooth change of distance.
    It's worth while talking in terms of Calculus. An abrupt (discontinuous) change in a variable can result in an abrupt change in the rate of change in the (continuous) value another variable. Integrating a step change will produce a change in slope of the integral.
  7. Jan 3, 2017 #6


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    An instantaneous change in acceleration requires a force law with a discontinuity. In most cases, such force laws are only approximations. If you look closely enough you'll eventually hit the quantum world and discover that position is not only not always twice continuously differentiable. It is not even a well defined function of time.
  8. Jan 3, 2017 #7


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    In the kinematics of collisions, it is a good ruse to talk in terms of an Impulse. You ignore the fact that the collision has to have taken a finite time and you define Impulse as Force times time it's applied for. Given a 'before' situation, you can apply a given Impulse (any combination of forces times time) and produce the same result. If it's a truly elastic collision the same thing will happen to a steel ball bearing or a plastic 'superball' or even a very soft sponge (remembering that only an ideal sponge can be perfectly elastic)
    It is very common to sidestep the problem of discontinuities in the Maths of Science.
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