Uncontinuous Motion: Nature's Examples

[symplectic_manifold]

Discontinuous motion

I've got a short curiosity question:
Are there in nature examples for discontinuos motion, that is when the functions a(t), v(t), x(t) are not continuous? Is there anything of this sort?

 

[Claude Bile]

Discontinuities in x(t) imply infinite velocity (impossible).
Discontinuities in v(t) imply infinite force (impossible).
Discontinuities in a(t) occur all the time. The motion of such a body still remains continuous however.

Claude.

 

[vaishakh]

Discontinuities in a(t) occur all the time. The motion of such a body still remains continuous however.
Claude.

Physicists while giving equations always satisfy mathematical conditions. but they just approximate them when they cannot help to find the actual value. what you said above is suich an example. it is not mathematically correct that discontinuity occurs in a(t).

in that way there is dicontinuity even in velocity as well as displacement.
in projectile equation can there be an initial velocity to the object without acceleration if it is projected. if you say it is due to the movement of hands before throwing, then after the throw the acceleration which while throwing has a value has to reach zero.

much more familiar is the case of free fall. you may have seen questions like a balloon or elevator going up at a constant acceleratin and a stone being dropped from it. there we just find time by the second kinematical equation. we neglect the time taken for the acceleration to reach g from the balloon value.

 

[mathphys]

It depends

I will give an example of discontinuos velocity related to shock waves.
When a gas with initial velocity u0 is shocked its velocity changes drastically across the shock wave (surface of discontinuity) to u1. The velocity is discontinuos here for any time t (at the position of the shock), u0 in front of the shock and u1 behind it. The same applies for the density and other properties.

Of course that is because the width of the shock front is assumed to be depictable, i.e, it is modeled as a surface in 3D. In reality it has a short but finite length and severe fast (but continuos )changes occur there. Nevertheless, this is generally ignored because of the scale in time and space in which this changes occur. People working in this field deal everyday with discontinuos velocities, in fact, the hydrodynamical equations admit discontinuos solutions.

So at least as a mathematical way of modeling physical phenomena, yes discontinuos velocites are possible. Because this 'models reality' better than continuos approximations for certain cases. Until Today.

 

[dx]

we neglect the time taken for the acceleration to reach g from the balloon value.

Baloon value?