Suppose you throw a ball into the air. Do you think it takeslonger to reach its maximum height or to fall back to earth fromits maximum height? We will solve the problem in this project, butbefore getting started, think about that situation and make a guessbased on your physical intuition. 1. A ball with mass m isprojected vertically upward from the earth’s surface with apositive initial velocity v0. We assume the forces acting on theball are the force of gravity and a retarding force of airresistance with direction opposite to the direction of motion andwith magnitude p|v(t)|, where p is a positive constant and v(t) isthe velocity of the ball at time t. In both the ascent and thedescent, the total force acting on the ball is pv mg. (Duringascent, v(t) is positive and the resistance acts downward; duringdescent, v(t) is negative and the resistance acts upward.) So, byNewton’s Second Law, the equation of motion is mv0 = pv mg Solvethis 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 ithits 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 withmass 1 kg and initial velocity 20 m/s. Assume the air resistance is1 10 of the speed. 1 4. Let t2 be the time at which the ball fallsback to earth. For the particular ball in Problem 3, estimate t2 byusing a graph of the height function y(t). Which is faster, goingup or coming down? 5. In general, it’s not east to find t2 becauseit’s impossible to solve the equation y(t)=0 explicitly. We can,however, use an indirect method to determine whether ascent ordescent is faster: We determine whether y(2t1) is positive ornegative. 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 lnx is increasing for x > 1. Use this result to decide whethery(2t1) is positive or negative. What can you conclude? Is ascent ordescent faster?

Please answer #3

Please restate the problem to be solved, and define allvariables and parameters. Please explain your reasoning andstrategy for solving the problem. Please go over basic principalsor key processes underlying the problem that was addressed in thepaper. Please include an interpretation of the information in thecontext in which the problem was solved. Please state yourconclusions in complete sentences which stand on their own

Sorry, I know this is a lot