PLTL Revenge with the \"Tater Balloon Your 2.1 m tall physics professor jogs every day at a constant speed of 3.0 infs along a straight path that passes under a bridge. You need revenge for that last quiz. Here are your choices: 1 . [\\J You will drop a water balloon off the biidge, from rest, a height of 4. 5 meters above the ground. For convenience, take directly below, where the balloon would hit the ground, to be so = 0. Where, along the path, should the professor be when you release the balloon? . You realize that with the straight drop, he will clearly see you. So you decide instead to throw it horizontally with speed 15.0 1111's. Again, where should the professor be when you release the balloon? . At what angle above the horizontal would you need to throw the ball such that it hit the professor when he had reached 20.0 meters from directly below you? Assume the same speed of 15.0 mfs and break the velocity into its components, in terms of the angle from the horizontal. Where should the professor be when you make the throw? . Then, nally, you remember that when unjustly wetted, your professor can rLui really fast. So you would like to throw the balloon at an angle such that it hits its mark at the farthest possible point along the path. At what angle from the horizontal should you launch the balloon? Again, assume that your throw still has speed 15.0 mfs. This is a problem similar to the one above, except that you don't yet know the range. This is a problem of maximizing the range, and it is not 45, as in the case where a projectile starts and ends at the same elevation. You will need to use calculus techniques to nd this angle. Starting with the projectile equations, express the initial speed in the x and y directions in terms of the speed (v9) and angle (3). Next, eliminate time by solving for t from the x equation and substituting it into the y-equation. You will have a quadratic equation in it. Use the quadratic formula to express the solutions for x. The range, x, will be a function of the angle (sinB, cosB, etc.) and the constants v0, g, and the change in height. The angle that maximizes (and minimizes) this value can be found by di'erentiating x with respect to B, and setting that derivative to zero. And finally, where should the professor be when you release the balloon\