Express all answers in terms of variables given in the question and natural constants (like the acceleration due to gravity g) and/0r quantities dened thereby. Show all work for credit. 1) A veterinarian needs to tranquilize a monkey using a dart gun. The monkey is at a height h in a. tree, which is a distance d from the vet. The vet knows that as soon as they pull the trigger and release the tranquilizer dart at the monkey, the monkey will release their hold and drop from the tree. In this exercise you'll show that to successfully tranquilize the monkey, the vet should aim at the monkey (not above or below) . (i) The vet aims right at the monkey and pulls the trigger. The dart leaves the gun with speed no. How long does it take the dart to reach the location of the tree? (ii) How far above the ground is the monkey by the time the dart reaches the tree? (iii) Show that when the dart reaches the tree, it is at the same height as the monkey and therefore tranquilizes the monkey. (iv) If the vet wanted to tranquilize the monkey right as the monkey landed, what dart velocity 110 should the vet use? X h Vo Vo h d d adrag Figure 1: Diagrams for HW1 Problems.2) In skee ball the player rolls a ball with initial speed v0 up a ramp that makes an angle 6 with respect to the ground and that, at its end, has height h. As the ball goes up the ramp it slows down with an acceleration of magnitude a. The aim is to put the ball in a bucket, also at height h, but a distance d away from the ramp. The goal of this problem is to nd the value of on for a successful roll. Avoid using the \"Range Equation\". Rather use basic kinematic principles. (i) What is the speed of the ball of as it leaves the top of the ramp? (ii) What is the time t that the ball is in the air? (iii) How far does the ball travel in the +3: direction during its ight? (iv) For what value of to does the ball land in the bucket? 3) A particle with initial velocity on at time t = O in a viscous uid will slow down due to the drag force of the uid on the particle. The faster the particle moves, the stronger the drag. The resulting velocity of the particle will be v(t) : v0 exp(bt). (1) A word on notation: \"exp\" is the exponential function. It is the inverse of the natural logarithm and sometimes written exp(a:) = e\". (i) Compute the acceleration a(t) of the particle. (ii) Use the result from part (i) to show that the drag force produces an acceleration that is proportional and in opposition to the velocity. What is the constant of proportionality? (iii) Compute the displacement As(t) of the particle since if = 0. (iv) Even though the particle's velocity never fully reaches zero, it slows down fast enough that there is a limit to the particle's displacement. Use the result from part (iii) to nd the displacement limit that the particle approaches after a long time