-2

I have what appears to be a paradox of motion, but it could just be my misunderstanding. It works under the assumptions that motion is fundamentally continuous, that position, velocity, acceleration and all possible $n^{th}$ derivatives of position are continuous functions. If an object is at rest then all $n^{th}$ derivatives of motion are 0, then suddenly as the object moves it changes velocity, in order to change velocity it must accelerate, in order to accelerate it must jolt etc. All possible $n^{th}$ derivatives of motion become non-zero instantaneously. how then can motion be continuous?

Ryan Cole
  • 99
  • 8
    If an object is at rest then all nth derivatives of motion are 0 - An object being at rest only implies that the first derivative of position is $0$, not any of the others. – HDE 226868 Jul 29 '18 at 23:50
  • Possible duplicates: https://physics.stackexchange.com/q/111251/2451 and links therein. – Qmechanic Jul 30 '18 at 05:16

2 Answers2

3

Why would you assume that "all possible $n^{th}$ derivatives of position are continuous functions"? From Newton's $F = ma$, that would mean that forces have to have "all possible $n^{th}$ derivatives" also continuous, and there's no reason to believe that.

And, aside from the physics issues, that restriction doesn't, as a matter of mathematics, restrict against all motion. There are smooth non-trivial functions with all derivatives vanishing at zero.

Bob Jacobsen
  • 14,439
1

If an object is at rest then all nth derivatives of motion are 0

This is incorrect. Consider, for example, a ball thrown straight up in the air. At the top of the trajectory the velocity (the first derivative of the position) is zero; the ball is at rest. However, the second derivative is non-zero (it is $-9.8 \frac{m}{s^2}$). Indeed, the second derivative is non-zero (constant) throughout the motion.

Note also, that in this example the position, the velocity, the acceleration, and all higher derivatives are continuous functions. The position is a quadratic function of time, the velocity is linear, the acceleration is constant, and all higher derivatives are constant (and equal to zero).

hft
  • 19,536