or is everything in nature continuous?
How do I know if F is continuous or not?
The gravitational force exerted by the Earth on a unit mass at a distance r from the center of the planet is
where M is the mass of Earth, R is its radius, and G is the gravitational constant. Is F a continuous function of r?
BTW, if you can't read it, it is
and
Well, let's not be too hasty about that. Quantum field theory seems to imply that there is a "cut-off" level in the structure of space-time. i.e. It appears that space-time has a non-continuous structure at extremely small scales. (This is, at least, one of the more reasonable ways to remove the ultraviolet divergent integrals.)Originally Posted by ThePerfectHacker
For a more basic example, consider an object falling into water. The speed function is certainly continuous, but the acceleration function has a discontinuity at the water-air interface.
-Dan
The acceleration cannot be continuous. Before the object hits the water the magnitude of the acceleration is g, but after it is in the water the acceleration is g-B/m, where B is the bouyant force and m is the mass. Thus there is a discrete step in the acceleration function at the interface.Originally Posted by mooshazz
-Dan
Addendum: Apologies!
I was thinking about this again. I had taken the above argument from a problem I had seen worked out several years ago, but it didn't address the following issue: the bouyant force is increasing from zero from the time the object strikes the water until the time it is fully submerged. Thus the acceleration IS in fact continuous as mooshazz stated. It is presumably possible to have an object of such a shape that the bouyant force would be a function of time, but this would not hold true for a general object. Thus we can state that some derivative of the acceleration function will be discontinous, probably the first derivative for an ordinary object (that is to say an object with the typically random shape found in nature).
-Dan