4 Cannonball Bonanza A cannonball is shot straight up with an initial speed v. It reaches a maximum height, ymar and then falls back down. (a) Prove that the maximum height the cannonball reaches is proportional to the square of the inintal velocity. (b) If you double the initial velocity, by what factor will the maximum height change by? (c) Prove that the cannonball travels at a velocity of + when it has traveled half way up and when it has traveled half way back down. (d) Does it take more time for the cannonball to go from where it is shot to half way up, or more time for the cannonball to go from its max height to half way back down? Use the facts from part c and the concept of average speed in your argument. (e) Prove that the cannonball takes a time of t - (22) to rise to half of its max height after being shot. (f) Prove that the cannonball takes a time of t to get to its max height after being shot. (g) Prove that when the cannonball is halfway back down the clock reads: t = (2+2), assuming the clock started when the cannonball was shot. (h) When the first cannonball is halfway back down from its max height, you want a second cannon ball, launced after the first, to collide with it. However, you must fire the second cannonball at the correct clock reading, or else the collision won't occur half way up. Assume both cannonballs are shot up at the same speed v. Prove that the clock reading the second ball should be launched at is t 2, assuming the clock started when the first cannonball was shot. 9 Physics Problems Page 2 ONE DIMENSIONAL KINEMATICS Mr. Keye (i) Calculate the height of the first cannonball when the second one is shot. Express your result as a factor (in decimal form) of ymar. The factor will be