Question

Suppose you throw a ball into the air. Do you think it takes longer to reach its maximum height or to fall back to earth from its maximum height? We will solve the problem in this project, but before getting started, think about that situation and make a guess based on your physical intuition. 1. A ball with mass m is projected vertically upward from the earth’s surface with a positive initial velocity v0. We assume the forces acting on the ball are the force of gravity and a retarding force of air resistance with direction opposite to the direction of motion and with magnitude p|v(t)|, where p is a positive constant and v(t) is the velocity of the ball at time t. In both the ascent and the descent, the total force acting on the ball is pv mg. (During ascent, v(t) is positive and the resistance acts downward; during descent, v(t) is negative and the resistance acts upward.) So, by Newton’s Second Law, the equation of motion is mv0 = pv mg Solve this di↵erential equation to show that the velocity is v(t) = ✓ v0 + mg p ◆ ept/m mg p 2. Show that the height of the ball, until it hits the ground, is y(t) = ✓ v0 + mg p ◆ m p ⇣ 1 ept/m⌘ mgt p 3. Let t1 be the time that the ball takes to reach its maximum height. Show that t1 = m p ln ✓mg + pv0 mg ◆ Find this time for a ball with mass 1 kg and initial velocity 20 m/s. Assume the air resistance is 1 10 of the speed. 1 4. Let t2 be the time at which the ball falls back to earth. For the particular ball in Problem 3, estimate t2 by using a graph of the height function y(t). Which is faster, going up or coming down? 5. In general, it’s not east to find t2 because it’s impossible to solve the equation y(t)=0 explicitly. We can, however, use an indirect method to determine whether ascent or descent is faster: We determine whether y(2t1) is positive or negative. Show that y(2t1) = m2g p2 ✓ x 1 x 2 ln x ◆ where x = ept1/m. Then show that x > 1 and the function f(x) = x 1 x 2 ln x is increasing for x > 1. Use this result to decide whether y(2t1) is positive or negative. What can you conclude? Is ascent or descent faster?

Please answer #3

Please restate the problem to be solved, and define all variables and parameters. Please explain your reasoning and strategy for solving the problem. Please go over basic principals or key processes underlying the problem that was addressed in the paper. Please include an interpretation of the information in the context in which the problem was solved. Please state your conclusions in complete sentences which stand on their own

Sorry, I know this is a lot

Answer 1