2 FREELY FALLING OBJECTS - we will consider the case where objects move in a gravity field – namely free-fall motion. We will neglect [for a time] air resistance on an object.
3 Galileo's Observation Galileo did more than just observe that spheres of different mass struck the ground at the same time when dropped from rest, at the same time, from a balcony of the Leaning Tower of Pisa. This observation, alone, allowed him to conclude that the spheres fell with the same acceleration, independent of the mass they had.
4 Galileo's Observation Galileo further noted that when a sphere was released from rest, it traveled distances, in successive equal time intervals, that were in the sequence of the odd integers. What is meant by this is that if the sphere falls 1 step in the first time interval, then in the second equal time interval, it falls 3 steps. In the third such equal time interval, it falls 5 steps, and so on. Thus, the sequence of successive distances fallen by the sphere is 1, 3, 5, 7, 9, 11, etc., steps.
6 Free Fall On Earth, when an object falls under the influence of gravity, it speeds up, or accelerates. Thus, an object doesn’t fall at a constant speed. Its speed increases at a constant rate though.
7 FREELY FALLING OBJECTS All objects feel a force acting on them directed toward the center of the earth. This force causes the object to accelerate at a constant or uniform manner. The symbol [g] will be used for this acceleration and the magnitude of this acceleration is 32 ft/s 2 or 9.8 m/s 2 OR 10 m/s 2
8 Diving You jump off a very high diving board into a pool. How do you speed up? t = 0 sec v = 0m/s t = 1 sec v = 9.8m/s t = 2 sec v = 19.6m/s t = 3 sec v = 29.4m/s
9 Direction?? & How Far??? Upward is usually considered the positive direction. Downward is usually considered the negative direction. Thus, the acceleration from gravity is sometimes given a negative sign. g = - 9.8 m/s 2 or –32 ft/s 2 How fast you speed up doesn’t necessarily indicate how much farther you will go. The distance you cover is proportional to the square of the time. d = ½ gt 2
10 The ball begins with a positive upward velocity. Due to the acceleration from gravity, the velocity is gradually reduced to zero at the top. The ball continues down, with an increasing negative velocity.
11 Ex: If a stone falls off a cliff for 1 sec, how far does it fall? For 2 sec? GIVEN: v i = 0 m/s; a= -9.8 m/s 2 ; t = 1 s UNKNOWN: d = ? EQUATION : d = v i t + ½ gt 2 SOLVE : d = v i t + ½ gt 2 SUBSTITUTION: d = v i t + ½ gt 2 d = ½ (-9.8m/s 2 )(1s) 2 d = -4.9m d = ½ (-9.8m/s 2 )(2s) 2 d = -19.6m
12 SOME EXAMPLES 1. A pumpkin is dropped from the top of the school. The time it takes for the pumpkin to hit the ground is 2 seconds. At what speed does it hit the ground? GIVEN: v i = 0 m/s; a= -9.8 m/s 2 ; t = 2 s UNKNOWN: v f = ? EQUATION : v f = v i + at SOLVE : v f = v i + at SUBSTITUTION: v f = v i + at v f = 0 m/s + (-9. 8m/s 2 )(2s) = -19.6 m/s
13 2. A pumpkin is dropped from the top of the school. The time it takes for the pumpkin to hit the ground is 2 seconds. How tall is the school at that point? GIVEN: v i = 0 m/s; a= -9.8 m/s 2 ; t = 2 s UNKNOWN: d = ? EQUATION : d = v i t + ½ gt 2 SOLVE : d = v i t + ½ gt 2 SUBSTITUTION: d = v i t + ½ gt 2 d = ½ (-9.8m/s 2 )(2s) 2 d = -19.6m
14 Free Fall You might notice that some objects don’t seem to fall as fast as others. Ex: rock, feather The feather is more susceptible to resistance from air molecules, so it is slowed more. In the absence of air, a rock and a feather would fall at exactly the same rate. Where could you accomplish this?