The Mushroom Kingdom

As Mario navigates the Mushroom Kingdom he often has to jump towards falling mushrooms, undergoing projectile motion. Suppose that during one such adventure, Mario would like to catch a power-up mushroom that will begin falling from the edge of a platform which is 7.0 m in front of Mario and 13.0 m above Mario's head. For simplicity, we will neglect the horizontal motion of the mushroom as it leaves the platform and assume the mushroom begins to freefall roughly vertically when it reaches the edge. The acceleration due to gravity in the Mushroom Kingdom is roughly 1/5th of that on Earth, 1.96 m/s2 downwards. mario.jpg a) Show that if Mario begins to jump at the moment the mushroom leaves the platform and begins to freefall, Mario should angle his jump pointing at the edge of the platform, i.e., tan𝜃0=ℎ𝑑 , where h and d are the initial height and horizontal distances to the edge of the platform as shown above, in order to successfully catch the mushroom as it falls. b) If Mario catches the mushroom after it has fallen 7.0 m from its initial height, how fast was Mario moving as he jumped? c) How long are Mario and the mushroom in the air before they collide? 6. Two ladybugs land on a vinyl record of radius R = 21 cm which is initially at rest. The first ladybug lands halfway from the center of the disc, while the other lands on its edge as shown below. The disc then begins to rotate, speeding up with an angular acceleration given by 𝛼(𝑡)=𝛼∞(1−𝑒−Ω𝑡) , where 𝛼∞ undefined2undefinedΩ = 0.368 s-1. After 12.6 seconds, the record reaches its design speed and stops accelerating, continuing to rotate with a constant angular speed. ladybugs.jpg a) What are the final angular speeds of the ladybugs? b) How many times does the record turn before reaching its design speed? c) What are the magnitudes of their accelerations after 7.23 seconds have passed? (Are they different?)