Required The science base on a small moon has detected an incoming asteroid. The only things standing between you and fiery death is your calculus skill. Trouble Defense 3/ Cannon Science Base The gravitational constant on the moon is g = 2. There is a defense cannon at (0, 0) that launches projectiles at a fixed speed ||Ul| = 20 km/s. 100 km away, at (100, 0), is the science base. The science base has established that the asteroid has the following postion function. Ta(t) = -4t - 100 ya (t) = -8t - 400 The asteroid will hit at t = -50, i.e. 50 seconds before midnight tonight it will pass through (100, 0). You must find a firing solution, i.e. an angle 0 and a time tare at which to fire the projectile. The Defense Cannon Manual suggests the following step by step approach. (a) Pick an angle 0 and assume the projectile is launched at t = 0. Determine whether the curves of the asteroid and the projectile cross. Give the times to when the asteroid reaches that point and the time to when the projectile reaches that point. (The numbers will get messy, it's okay to use a calculator and decimal numbers.) REMINDER. If the projectile is fired from (0, 0) at t = 0 with initial velocity (v1, v2), then at the time t the position of the projectile is given by x(t) = vit, y(t) = vat _ gt- 2 . (You may remember this from a Mechanics course, and we will derive this in a lecture this week, but for now let's take it on faith.) (b) From what you found in part (a), determine the time tare at which the projectile should be fired to hit the asteroid at that point. Provide explanation (the cannon's AI will refuse solution with incomplete work.)