Question

By now, you may be getting sick of hearing that we may ignore air resistance" in problems. We did that before because we did not have knowledge of forces. But now... Let's see just how powerful air resistance is on Earth. Assuming that we are standing on the surface of Earth, let's find the fractional difference between the maximum heights reached for each case. 1. Simple case first: a ball thrown upwards without air. Find an expression for the maximum height reached if the ball is thrown vertically upwards. 2. Now, lets do it for real. The air exerts a force opposing motion in the form of, Farag=CpAv(t)? (1) is the drag coefficient and p is the density of the fluid that is being traveled where through a.) Draw a force diagram for the forces acting on the ball in mid-flight. b.) Find Fnet on the ball. c.) Find the resulting acceleration the ball undergoes. d.) Using this correct acceleration, find an expression for the maximum height the ball will reach. 3. Take the ratio of mar air. This represents what fraction of the theoretical max height you actually reach as a function of mass, instantaneous velocity, and shape of the object. (Note: The u in the friction term is NOT vo, it is a function v(t). This means that the force of friction dynamically adjusts depending on how fast an object is moving.) Debriefing Questions: - If the mass increases, how does this affect friction's influence? Will you get higher or lower than no air? Does this make sense? - To help visualize how this effect changes with velocity, go to www.wolframalpha.com and plot this function. It's super easy. Just input this into the query bar: plot y = 1/(1+x 2). This approximates our expression by simplifying all the constants" of the problem to 1. This approach allows us to isolate the specific function behavior we want to analyze. Notice how as speed increases it either direction, the the max height reached drops off rapidly. To check how mass affects the friction, alter the above equation in Wolfram by dividing the r2 by a number (say 2, then 10, then 100) and re-plotting it. Look carefully at how the axes change.

Answer 1