## Galileo Galilei

#### Exercises

1. An object in free-fall is accelerated 16 ft/sec2 by gravity.  If it takes exactly 4 seconds to fall to the ground, at what height was it dropped?  [Hint: Use the mean speed law.]
2. Amy is driving along when she notices her friend Bill's car in her rear-view mirror.  At the moment that Bill speeds past her at 60 mph = 88 ft/sec, Amy starts accelerating at a rate of 1 ft/sec per second from her initial speed of 50 mph = 73 ft/sec to catch up with him. (a) Draw a graph of their speeds as functions of time, with 0 seconds corresponding to the time at which Bill first passes Amy; according to the mean speed law, how long will it take for Amy to catch up with Bill?  (b) How fast will Amy be going when she overtakes Bill?  (c) How far away from the point that Bill passed her will she pass Bill?  (d) When, in this period of time, is Amy furthest behind Bill?
3. Suppose a rocket gains speed v (measured in m/sec) according to the formula v = 20t2 where t is the time after liftoff in seconds.  How far will the rocket travel in its first 10 seconds of flight? [Hint: write a suitable integral to answer the question.]
4. A cannonball is fired straight out from the top of the battlements of a castle onto the plain 100 ft beneath at a speed of 60 ft/sec.  Use the Galilean equations of motion for the cannonball (and g = 32 ft/sec2) to determine (a) how long it takes for the ball to fall to the ground, and (b) what the range of the shot is (that is, how far from the castle wall the ball strikes the ground).
5. In the situation of the previous problem, if the cannonball strikes the top of a seige tower exactly two seconds after it is fired, how tall is the tower?