Fastball (FB), a teenager who became a superhero after catching a radioactive baseball, can fly, and she can throw any object as fast as a jet aircraft with no air resistance at all. Right now FB is at rest, high above the ground, and her pet bird Supersparrow (SS) is circling well above her. FB wants to toss a donut up to SS.

(1) SS is 7.85 km above FB. Calculate the minimum initial velocity the donut would need in order for FB to throw the donut straight up to SS's altitude. Be careful with units! Check: does the calculated velocity give the correct displacement?

(2) Calculate how much time it would take the donut to reach SS's altitude if launched at this initial velocity. Check: put t back into the original expression, or a different one: do you get the right displacement?

(3) Before actually throwing the donut, FB realizes that SS is going to reach the point directly above her 12.5 s sooner than the time found in #2, so the donut has to be thrown faster than that to get to SS in time. Ex: if the donut would take 105 s to reach SS's altitude in #2, it now needs to get there in just 92.5 s, so you might choose actual time to rise: t = 88.4 s.

State the random actual time you've chosen, then calculate the actual initial velocity needed to make the donut reach SS on time. Tip: the donut won't stop when it reaches SS's altitude if thrown at a velocity faster than #1. Check: FB must throw the donut faster than the velocity calculated in #1! Fastball launches the donut at the velocity calculated in #3, but SS misses the catch. The donut rises to some forever unknown maximum height, then begins falling back down again.

(4) MVQ, RNG: SS chases the donut, catching up to it either when it's 3.0 km above Fastball (if the second-to[1]last digit of your phone number is even) or 3.12 km below Fastball (odd). State your assigned version and random: how many km above or below Fastball does SS catch the donut? No calculations.

(5) Make a sketch showing the interval from launch to interception, showing

• the donut at launch (beginning), and then just before interception (end)
• two vectors (arrows) labeled vi, vf, showing the donut's velocity at its initial and final positions
• one vector labeled a, showing the direction of the donut's acceleration during this interval
• one vector labeled x (or y) showing the donut's displacement for this interval The sketch should be at least as large as your hand, so that all parts of it are easy to read online.

(6) Still treating the entire flight of the donut, from launch (at the initial velocity found in #3) to just before interception, as a single interval of motion, calculate the donut's velocity the instant (before) SS catches it. This means not breaking the flight into sub-intervals. Checks: see Exam Info > Square roots. Given which direction the donut is traveling, what must be the sign of its velocity? And given where the donut is intercepted, relative to FB, should this final velocity be faster or slower than the initial velocity?

(7) Calculate the donut's total flight time from launch to interception, again not breaking the flight into sub[1]intervals and solving them separately. Check: compare the calculated flight time to the partial rise time chosen in #3: which should be larger?