EXAMPLES. III. 1. A point passes over 9 ft., 10 ft., i ft. and 12 ft. in 4 consecutive seconds; find its average velocity for the seconds. 2. Find the average velocity of the point in Question 1 (i) for the first 3 seconds, (ii) for the last 3 seconds. 3. A point passes over 20 yds., 24 yds., 28 yds., 32 yds. and 36 yds. in 5 consecutive seconds, shew that the average velocities for the 5 seconds, for the 3 middle seconds and for the I middle second are all equal. 4. A point has an average velocity of 2 velos, 3 velos, 4 velos, 5 velos, and 6 velos respectively in 5 consecutive intervals of 5 seconds each ; find the average velocity for the 25 seconds; how far does the point go in the 25 secs.? 5. A point moves over a ft., 2a ft., 3a ft. and 4a ft. respectively in 4 consecutive intervals of b seconds each; find its average velocity for the whole time. 6. A point has an average velocity of 130 velos, 11• velos, 9v velos, 70 velos and 5ð velos respectively in 5 consecutive intervals of t seconds each ; find the average velocity for the 5t seconds. 7. A point has an average velocity of 3 velos for 2 secs., 6 velos for the next 4 secs., 10 velos for the next 4 secs. and 15 velos for the next 6 secs.; shew that in the 16 seconds it passes over the same distance as a point which has an average velocity of 3, 5, 7, 9, 11, 13, 15, 17 velos respectively in 8 consecutive intervals of 2 seconds each. VELOCITY WHICH IS NOT UNIFORM. 9. Consider a point in motion not moving uniformly. Its average velocity during any interval varies directly as the distance passed over in that interval and inversely as the length of the interval. Its average velocity will be different for different intervals. It is the average velocity of a train for a certain interval, which is found by noting the number of seconds occupied by the train in passing successive quarter-mile posts; but it is not the average velocity which is meant when we speak of the velocity of a point at a certain instant. Consider two very long trains moving on parallel lines in the same direction. Let one of the trains (P) be moving with a constant velocity, say, 20 velos; let the other train (Q) start from rest and move gradually faster and faster. A passenger in train P will see his train gain on train Q at first; but as time goes on, the train P goes faster, until at last there comes an instant at which the two trains are moving exactly together and neither is gaining on the other; this is only for an instant; it occurs at a certain instant. The train ( immediately begins to gain on the train P; but there is a particular instant at which neither train is gaining on the other and the two trains are relatively at rest. At that instant we say, that the velocity of the train Q is equal to the velocity of the train P; that is, the velocity of the train ( at that instant is 20 velos; hence, DEF. The velocity of a moving point (which is not moving uniformly) at a given instant, is the velocity of another point which, moving with uniform velocity in the same direction, is at that instant at rest relatively to the first point. IO. UNIFORMLY INCREASING VELOCITY. II. I 2. The simplest case of a point moving with velocity which is not uniform is that of a point whose velocity steadily increases or whose velocity steadily decreases. When a point is said to have velocity which is increasing or decreasing, the point does not move for an interval, with uniform velocity and then suddenly move with a different velocity, but the velocity at any instant is the velocity which gradually increases or decreases. DEF. A point is said to move with uniformly increasing velocity when in the course of each of a series of equal intervals its velocity receives an equal increment, no matter how large or how small these equal intervals may be. The velocity grows steadily ; it does not increase by jerks or steps. Suppose that a point Pis moving with uniformly increasing velocity; suppose that at a certain instant its velocity is 100 velos, and that after an interval of 1 second its velocity is 101 velos; then at the end of one more second its velocity will be 102 velos; at the end of another halfsecond its velocity will be 1025; and so on. Its velocity increases at the rate of i velo per second. Example. A point moving with uniformly increasing velocity has velocities 20 velos and 28 velos respectively at the beginning and at the end of an interval of 12 seconds, find its velocity 3 seconds before the beginning of the interval and 3 seconds after the end of the interval. In 12 seconds the velocity increases by 8 velos; therefore in 1 second the velocity increases by js, or s velo. Hence, 3 seconds before the instant at which it has 20 velos it has (20 – 3 xs) velos, or 18 velos. And 3 seconds after the instant at which it has 28 velos, it has (28+3 x 3) velos, or 30 velos. 13. A similar definition applies to the expression uniformly decreasing velocity. Example. A point moving with uniformly decreasing velocity has 30 velos; after 5 seconds it has 20 velos; when will it come to rest? The velocity diminishes by 10 velos in 5 seconds, therefore it diminishes by 30 velos in 15 seconds, When it has diminished by 30 velos it has o velo and is at rest. Therefore it comes to rest in 15 secs. EXAMPLES. IV. In the following examples the motion is in each case that of uniformly increasing or decreasing velocity. 4 1. A point has 8 velos at the beginning and 9 velos at the end of a certain second ; how many velos has it after more seconds ? 2. In a certain interval of half a minute the velocity of a point increases from 10 velos to 100 velos; what was its velocity at the middle of the interval ? 3. In 1 hour the velocity of a point decreases from 300 velos to 120 velos; what was its velocity at the end of each quarter of that hour? 4. At noon a point is moving with 20 velos; at 4 P.M. it has 100 velos; what velocity has it at 2.15 P.M. ? 5. At 2 o'clock a point has 7 velos; at 2.45 it has 142 velos; what has it at 3.30 ? 6. A train is being pulled up, and moves with uniformly decreasing velocity; at a certain time it is going 60 miles per hour, after 18 seconds it has 80 velos; when will it stop? 7. A point which started from rest has after 4 seconds a velocity of 100 yds. per minute ; when will it have so velos? and when will it be going at the rate of 60 miles an hour? 8. A point at a certain time has ii velos; 2 minutes later it is moving with a velocity of 30 miles an hour; when was it at rest ? 9. A point has 128 velos and its velocity is decreasing at the rate of 32 velos per second; when will it come to rest ? and how long will it be before it again has 128 velos? 10. A point moving at the rate of 11 ft. per second has its velocity increased by 7 ft. per second per second ; when will it be moving at the rate of 60 miles an hour? 11. A point whose velocity is decreasing at the rate of (32 ft. per second) per second comes to rest after 3 seconds ; what was its velocity at the beginning of the 3 seconds ? and what at the middle of the 3 seconds ? and what was it after 3 seconds more? 12. A point whose velocity is increasing at the rate of 32 ft. per second per second is moving vertically downwards with a velocity of 112 ft. per second; when was it at rest? and when was it moving vertically upwards with a velocity of 112 ft. per second? 14. When a point is moving with uniformly increasing velocity, its average velocity in a given interval of time is greater than the velocity which the point has at the beginning of the interval and is less than the velocity which the point has at the end of the interval. For its velocity at the beginning of the interval is less than its velocity at any other instant in the interval, and its velocity at the end of the interval is greater than its velocity at any other instant in the interval. 15. The same thing may also be stated thus; Let a point, moving with uniformly increasing velocity, at the beginning of a certain interval of t seconds have u velos, and let the point at the end of the interval have v velos; then, the distance passed over by the point in this particular interval of t seconds is greater than txu feet and is less than t x v feet. Example i. A point has 8 velos and its velocity is increasing at the rate of 2 velos per second ; shew that in the next four seconds it passes over more than 44 feet and less than 52 feet. Its velocity at the beginning of each second is 8, 10, 12, 14 velos respectively; therefore it passes over more than (8+10+12+14) feet in these 4 seconds. Its velocity at the end of each second is 10, 12, 14, 16 velos respectively; therefore it passes over less than (10+12+14+16) feet in these 4 seconds. Example ii. Shew by taking the velocity of the point in Example i at the beginning and at the end of each quarter second, that the distance passed over in the 4 seconds lies between 47 feet and 49 feet. At the beginning of each 4 second the velocities are 8, 81, 9, 99, 10, 10 etc.... up to 153, velos, respectively, and the distance passed over is therefore greater than 4 {8+81+9+93+. +15} feet, that is, greater than 47 feet. At the end of each 4 second the velocities are 87, 9, 9% etc.... up to 16, velos, respectively, and the distance passed over is therefore less than 4 {83 +9+93 + +16} feet, that is, less than 49 feet. Q. E. D. EXAMPLES. V. The following velocities are each uniformly increasing or decreasing. 1. A point has 8 velos and its velocity is increasing uniformly at the rate of i velo per second ; shew by considering the velocity at the beginning and at the end (i) of each second, that the distance passed over in the next 10 seconds is greater than 125 ft. and less than 135 ft. (ii) of each oth of a second, that the distance passed over in the 10 seconds is greater than 1293 ft. and less than 130 ft. 2. A point has 120 velos and its velocity is decreasing at the rate of 32 velos per second ; shew by considering the velocity at the end of each 34 th of a second that the distance passed over in the next second is less than 1041 ft. and greater than 1033 ft. 3. A point has 16 velos and its velocity is increasing at the rate of 32 velos per second; shew hy dividing the interval of time into ths of a second that the distance passed over in the next 3 seconds cannot differ from 192 feet by more than 18 inches. |