SECTION I. RECTILINEAR MOTION. CHAPTER I. VELOCITY, ACCELERATION. The distance of a point P from a point A is represented in direction and magnitude by the straight line joining A to P. The point P is said to be in motion relative to the other point A, when the distance of P from A is changing. The distance of P from A may be changing in length, or it may be changing in direction, or it may be changing both in length and in direction. While the direction of the distance of P from A remains unchanged the path of P is the straight line AP. We shall in this section discuss the motion of a point which moves always in a straight line. Accordingly, for the present, we are not concerned with any change of direction in the distances considered. Р H B Draw a straight line AB; let this line extend any length beyond A and beyond B. Let a point P be in motion relative to A in the line AB. Then the distance AP is different at different instants. Thus, suppose the point P at one instant to be at H; at another, at K; at another, at L; H, K and L being points whose distances from A do not change ; then, P being in motion relative to A, P alters its distance from A by the distance HK in the course of a certain interval of time. This is what is meant when it is said that P passes over the distance HK in a certain interval of time. L. D. I 2. We choose a foot as our unit distance. In what follows we shall frequently use the word interval as an abbreviation for interval of time. We choose a second as our unit interval. Let the distances from A of the above moving point P be observed at the ends of a series of equal intervals of time. Then if its motion is of such kind that in each of those equal intervals the point P passes over equal distances, and if this is true whatever be the length of these equal intervals, then we say that the point Pis moving uniformly; hence, 3. DEFINITION. The motion of a point P relative to another point A is said to be uniform when it is such that the point P passes over equal distances in equal intervals of time, however short those intervals are. Let us suppose that the point P is moving uniformly in the straight line AP. By its velocity we mean something which is doubled, trebled, etc. if the distance which it passes over in a given interval is doubled, trebled, etc., and which is halved, divided by three, etc. if the interval occupied in passing over a given distance is doubled, trebled, etc.; in other words, 4. DEF. The velocity of a point which is moving uniformly in a straight line is that which varies † directly as the distance passed over in a given interval, and which varies inversely as the interval occupied in passing over a given distance. Velocity is therefore something which we observe in a moving point. It is capable of measurement; for we can say that the velocity of a point is double, or that it is half, or that it is some multiple, of the velocity of another point. We therefore choose a certain velocity as our unit velocity, and give it a name; just as we choose a unit of length and call it a foot; and a unit interval and call it a second...; hence 5. DEF. We choose as our unit velocity the velocity of a point which moving uniformly passes over the distance I foot in the interval 1 second. + See Lock's Arithmetic, p. 162. We shall call this unit velocity a velo. A velo in a given direction, is a foot per second in that direction. The word per here denotes for every. NOTE. It is most important to notice that a point which has velocity requires an interval of time in which to pass over a finite distance. A point P may have velocity at a certain instant, but unless the velocity continues during some interval after that instant, the point does not pass over any finite distance. For example, a bullet which strikes a target is said to have a certain velocity at the instant at which it strikes. Example i. A point which moves at the rate of 3 feet per second has a velocity 3 times as great as that of a point which moves at the rate of 1 foot per second. Hence its velocity is 3 velos. Example ii. Express in velos the velocity of a point which moves at the rate of 60 miles per hour. This point in 1 hour passes over 60 miles; that is, in 60 x 60 seconds it passes over 60 x 1760 x 3 feet; but it passes over an equal distance in each second; бо х 17бо х3 therefore in i second it passes over feet; that is, 88 feet. бох бо Hence, its velocity is 88 velos. Example iii. A point has the velocity 8 velos; how many miles does it pass over in an hour ? The point has 8 velos; therefore, in 1 second it passes over 8 feet; and it passes over equal distances in equal intervals; therefore in 60 x 60 seconds it passes over 60 x 60 x 8 feet; 8х бох бо that is, miles; or, 57 miles. 17бох3 EXAMPLES. I. In each of the following questions the velocity given is uniform. 1. If I run 100 yards in 11 seconds, what is my velocity in velos ? 2. A man runs at the rate of 7 velos; how long would he take to run a mile with this velocity ? 3. Express in miles per hour (i) 40 velos, (ii) 100 yds. per minute. 4. How many velos has the extremity (i) of the minute hand of a clock which is i foot long, (ii) of the hour hand which is to in. long. [The circumference of a circle=42 of the diameter.] 5. How many velos has a man standing at the equator, in consequence of the daily revolution of the earth about its axis, the diameter of the earth being, say, 8000 miles? 6. The velocity of sound is 1118 velos; how long does it take to travel 13 miles ? 7. Which is the greater velocity, 60 miles an hour or 500 yards in 11 seconds ? •8. Find the ratio of the velocities 23 miles in 3 minutes and 10 ft. in a quarter of a second. 9. The following are the records' in foot racing (i) 100 yds. in 10 secs. (ii) a quarter of a mile in 49 secs. (iii) a mile in 4 min. 12 secs. (iv) ten miles in 51 min. 6 secs.; express the average velocity of each in velos. 10. The following are the records in bicycle racing (i) a mile in 2 min. 30 secs. (ii) 20 miles in 59 min. 6 secs. (iii) 22 miles 150 yds. in 1 hour; express the average velocity of each in velos and in miles per hour. 6. A point which has v velos passes over v feet in I second; it therefore passes over (t times v) feet in t seconds. Hence, if s be the number of feet passed over in t seconds by a point which has a uniform velocity of v velos, we have s=vt. Example. A point passes uniformly over a miles in b hours; express its velocity in velos. In bx 60 x 60 seconds it passes over a x 1760 x 3 feet, therefore 17бо х3ха in 1 second it passes over бох бохь feet; 22a feet; that is, 156 22a therefore it has velos. 156 7. The measure of the velocity of a point (when a velo is the unit) is the number of velos which it has. This is the ratio of the number of feet ed over in any interval to the number of seconds in that interval. Hence, the velocity of a point is measured by the rate at which it passes over distance per unit interval of time; in other words, by the number of units of length which it passes over in unit interval. EXAMPLES. II. The velocities given are uniform. 1. Express in velos a velocity of c miles per hour. 2. A point has v velos; how far does it go in h hours? 3. A point has k velos; how long does it take to go m yards ? 4. A point goes m yards in t seconds; how many miles does it go in h hours supposing its velocity uniform ? 5. A point goes 1 inches in 1 minute ; how far does it go in k days at the same rate ? 6. A point goes k feet in t seconds; how long does it take to go m miles with the same velocity? 7. A train goes n miles in k hours; how far does it go in t seconds? AVERAGE VELOCITY. 8. DEF. The average velocity of a moving point during a given interval, (in which it is not moving uniformly), is the velocity of another point which, moving uniformly, passes over the same distance in the same interval. Example. A point moves over 6 ft., 7 ft., 8 ft. and 9 ft. respectively, in four consecutive seconds; find its average velocity for the four seconds. In 4 seconds it passes over (6+7+8+9) ft. ; that is, 30 ft. Thus, in 4 seconds a point having the required average velocity passes over 30 feet; therefore in i second this point passes over 74 feet; that is, the required average velocity is 7} velos. |