You are standing at the corner in the rain, waiting for the light to turn green. As you wait, you start to spin your umbrella around so that its edge travels around a horizontal circle at a speed of one complete rotation every 3 seconds. If you look up at the umbrella, it spins counterclockwise. Your umbrella has a radius of 25 inches, and it's shaft is 9 inches long. You hold the umbrella 50 inches above the ground and stand facing the x-direction. You are going to find the radius of the splash of the water that is spinning off of your umbrella by looking at what happens when a drop of water flies off of it rim directly in front of you (and assuming the same thing happens everywhere else on the umbrella). I. The Initial Conditions Assuming that the water on the edge of the umbrella is motionless with respect to the spinning surface of the umbrella when it leaves the fabric, write the initial conditions of a drop of water that flies horizontally off the umbrella directly in front of you. (a) What is the initial position of the water drop? Ti = 25 in x+ 0in y+ 59 in (b) What is the initial velocity of the water drop? = [-41.89 in/s x x + 16.67 in/s x y + 0in's II. Integrate to Find the Drop's Trajectory After setting up the acceleration function, integrate up to the position. (a) What is the acceleration function of the water drop? a(t) = 18 + ly + 12 (b) What is the initial position of the water drop? "(t) = t ) * + It) y + t) 2 (c) What is the trajectory of the water drop? F(t) = It + 12 ) x + 7t + 1: 2 ) 9 + It + III. Find the Splash Radius Now use the trajectory to find how far away someone has to stand to not be hit by the water flying off of you spinning umbrella. (a) How long does it take to the drop of water to hit the ground? t = (b) What is the position where the water drop hits the ground? It = X + (c) What is the minimum safe radius for a bystander? D = 54.61 in JX