So in a physics lab, we threw a ball upwards and recorded its acceleration. If you look at the graph there is actually a period of time where it goes from positive acceleration to -9.8 m/s2. But how is that possible? Before I let go of the ball, it experiences a positive acceleration because I am applying a force causing the net force to be up, but as soon as I let go, the only force acting on it is gravity, so wouldn't the acceleration have to jump from some positive number to -g?
The acceleration does not have to be a continuous function of time. It can change discontinuously. Only the velocity and displacement need to be continuous functions of time.
I would imagine that as the ball leaves the hand, the upward force on it goes to zero very quickly, but not instantaneously - the hand isn't rigid. So, the acceleration will change very quickly from positive to negative, but not instantaneously.
There is nothing that says that the acceleration of an object can't be continuous. But there is nothing that says that the acceleration of an object cannot be discontinuous either.
If you model the hand as a rigid body, then I can see that the force from it would drop to zero instantly giving a discontinuous acceleration. But this is a consequence of our idealised model. I don't see how the force/acceleration could be discontinuous in a real situation.
There's a disparity between the mathematical model and reality, of course. I personally believe acceleration, jerk, jounce, etc are continuous in reality. Don't forget that the forces repelling the molecules between the ball and hand are actually 'theoretically infinite fields'.
Yes, agreed. But at this point, I think we are beginning to split hairs. When we analyze these problems, we are also neglecting relativistic effects, but that doesn't bother us much. And what about QM effects? You won't get an inaccurate answer, provided you understand the nature and potential error of your idealized formulation.
I concur with this as well. If a model is "accurate enough" then it's serving its purpose. I wasn't speaking from a standpoint of analysis as much as a philosophical one...
So at what acceleration does the ball enter free fall? My TA said when the acceleration becomes negative. Maybe because at zero acceleration there is no net force?
Hmm, this goes to the heart of what Chestermiller said was "splitting hairs". An object is considered to be in free fall if no forces are acting on it beyond the "pseudo-force" of gravity. The point at which the acceleration on the ball is zero and immediately thereafter becomes negative would occur (in a non-idealized world) while the ball was still in contact with the hand. [To be clear here: this would be in the very last moments of contact with the hand, as the elasticity of the skin was a contributing factor] At what acceleration does the ball enter free fall? That's easy: by definition, free fall occurs when only gravity is affecting it, and you already know that answer.