I am trying to set up a model using calculus in physics and I need assistance?

[zeromodz]

Okay, you know how the acceleration of gravity changes by a factor of r^2 when you are falling downward from the sky to the ground. Say I drop a ball from b = 50 meters to a = 0 meters(surface of the earth). The change in acceleration over time is called a "jerk" and its the second derivative of velocity.

So how could we calculate the actual final speed of the ball when it hits the surface of the Earth including instantaneous acceleration? I know that it is negligible, but I want to see if its possible to really calculate excluding air resistance.

I have an idea on how to start how to work this out, but I got a little confused because I am a little new to calculus and am not completely sure my work was correct. Using my intuition that a model of this would already exist if it was possible to make by other physicists.

 

[Curl]

Dude this is really easy, write down the function of the Earth's field and calculate the path integral between two points. Since the Earth's field is conservative (curl = 0) then the integral is path independent so you can evaluate the potential function at the end points and you get the same answer.

Or take a shortcut and write down the force of gravity on your object as a function of distance from the Earth's center, and integrate over your falling distance. Once you find the total work done on your object you can use Einstein's equations from special relativity to calculate the velocity from Earth's reference frame (since you are so picky about getting an accurate model, otherwise you'd use v=root(2gh) ). Acceleration is F/m, you should know that.