at, the space passed over in (t − 1) 1)2. Hence the space passed ing by Formula 4, S = = s = S― S' ata(t 1)2 = a(2 t − 1); i.e., s = 1⁄2 a(2 t − 1). (6) It is sometimes convenient to express velocity and acceleration in symbols. Thus a velocity of twenty centimeters per second may cm. sec. be expressed as 20 and the speed of a bicycle rider going at the ft. rate of a mile in four minutes is a speed of 22 that is, twenty-two feet per second. In the same way an acceleration of fifteen centimeters per second per second may be written 15 and an accelerasec. 29 ft. tion of twelve feet per second per second may be written 12 cm. sec.2 sec. 39. Effect of a Constant Force. Whenever a body is moving under the influence of a constant force only and there is no change in the resistance, the resulting motion is uniformly accelerated. The constant force with which we are most familiar is the force of gravity; hence, as an illustration of the effect of constant forces, we study the motion of a falling body. 40. A Freely Falling Body; Resistance of the Air. — A body that is moving under the influence of gravity alone is a freely falling body. This condition can be obtained only in a vacuum, as the air constantly offers a resistance to the passage of any body through it. Demonstrations. Trim a piece of stiff paper and a cork until each has the same weight as a shot or a bicycle ball. Drop all three from the same height at exactly the same time, and notice when they strike the floor. Since they all have the same weight, the force tending to give them motion is the same, but as they present dif ferent amounts of surface to the air, the resistance of the air varies. If the sheet of paper is let fall when it is flat, it will slide down on the air in various directions, but if a small part of its width is turned up at an angle, it will fall very steadily. Drop two balls of the same size, one of brass and one of wood, from a height of 20 ft. or more. They reach the ground at practically the same time. Why? These demonstrations indicate in a simple way the method used by Galileo to settle by experiment the discussion which had been vigorously carried on between those who thought that the velocity of a freely falling body was proportional to its weight and those who, like Galileo, thought it was the same for all bodies. The experiment was made by dropping various bodies from the top of the leaning tower of Pisa, and showed that Galileo was right. The effect of the resistance of the air upon falling rain is so great that drops from a high cloud have no greater velocity when they reach the ground than those from a cloud much nearer the earth. The effect of this resistance upon a small stream, falling from a great height, is to break the water up into a fine spray. This is well shown in the Upper Fall of the Yosemite, where the water reaches the foot of the fall as a fine rain or mist. ― 41. Measuring the Velocity of Falling Bodies. — (a) The Direct Method consists of dropping a small ball of some heavy material from the top of a tower like a shot tower and determining by actual measurement where it strikes a support at the end of the first second, second second, etc. One of the difficulties connected with this method is the height of tower required, since for a fall of 3 sec. the tower would need to be about 145 ft. high. (b) Galileo's Method. In all other methods the velocity of the falling body is reduced in some way. Galileo ac complished this by letting a ball roll down an inclined plane. If the length of the plane is made great in comA parison with the height, the ball will roll down the plane far more slowly C than it would fall in a vertical direction. B FIG. 13 If the experiment is carefully made, the results will be such as are shown in Table A, since the resistance of the air is slight. Let A1 (Fig. 13), the space passed over in the first second, be called d. Then it will be found that A2, the space passed over in 2 seconds, is 4 times as great, or 4 d; that A3, the space passed over in 3 seconds, is 9 times as great, or 9 d, etc., no matter what the proportional height of the plane is. These results are shown in the fourth column of Table A, from which are found the spaces passed over in the different seconds, as shown in the third col umn. Since the force is a constant one (it is certain fraction of the weight of the ball), the acceleration is constant, and is twice the distance passed over in the first second, or 2 d, per second per second. Notice that this is also the difference between any two successive values in column 3. The acceleration 2 d in turn gives the values in the second column. Time in seconds 1 2 3 4 TABLE A Velocity per sec. 2 d 4 d 6 d 8d Space passed over d 3 d 5 d 7 d Whole space passed over d 4 d 9 d 16 d By increasing the proportional height of the plane the velocity of the ball is increased, until, when the plane becomes vertical, the ball is no longer a rolling but a falling body and the acceleration, 2 d, equals g. Replacing a in Formulas 3, 6, and 4 by g, we have the formulas for falling bodies: v = gt, s = 1⁄2 g(2 t − 1), gt2. The value of g varies at different places on the earth, from about 978 cm. at the equator to about 983 cm. at the poles. At New York its value is 980.2 cm., or 32.16 feet. By making this substitution, these formulas may be written: FOR RESULTS IN FEET FOR RESULTS IN CENTIMETERS v = 32.16 t s = or v = 980.2 t, = 16.08(2 t S = 16.08 t2 These formulas for freely falling bodies are very important, and should be familiar to every student. S = 42. Graphical Analysis of a Falling Body. - The motion of a falling body can be analyzed graphically as in Fig. 14. Draw a vertical line and take a certain distance AB, from the top of the line A, as the distance the body falls the first second, equal to g. Measure from B twice this distance to represent the velocity S for 1 sec. 1 (} g) g S for 2 sec. 4 (} 9) v=2g S for 3 sec.-9 (19) A E (7) (8) (9) s in 1st sec. 1 (} g) s in 2d sec. 3 (} g) s in 3d sec.-5 († 9) in 4th sec. 7 (} 9) s in 5th sec. 9 (} g) G s in 6th sec. 11 (} 9) |