188 



THE POPULAR EDUCATOR. 



Fig. 100. 



down the incline D E ; and since the velocity of a falling body, 

 as we have seen, depends upon the vertical distance passed 

 over, its velocity is all along greater. 

 The space passed over is, however, 

 greater too ; but this is more than 

 compensated for by the increased 

 velocity. The curve of shortest 

 descent of all is found to be that 

 which has the greatest curvature 

 without rising as it approaches E. 

 If a pencil be fixed so as to project 

 horizontally from the rim of a wheel, 

 and made to trace a curve on paper 

 while the wheel is rolling on, it will 

 be exactly that of shortest descent. 

 As we shall see further on, there are other remarkable and im- 

 portant properties possessed by this curve, which is called a 

 Cycloid. (See Lessons in Geometry, XXIII., p. 309.) 



PROJECTILES. 



Having thus seen the laws which govern the motion of falling 

 bodies, we pass on naturally to notice the movements of pro- 

 jectiles. Here, of course, as before, the resistance of the air 

 impedes motion to a greater or less extent. This resistance 

 increases as the square of the velocity, for if the speed of a 

 body be doubled, it not only has to displace twice the bulk of 

 air, but it must do it with twice the velocity, and for this a four- 

 fold force is needed. As, however, our calculations would be 

 much complicated if we took this into consideration, we will 

 neglect it, but we must remember to make allowance for it in 

 our results. 



The path of a projectile is in a curve called a parabola, that 

 is, a curve similar to the one which we should obtain if we 

 were to cut a cone in a direction parallel to one side. (See Les- 

 sons in Geometry, XXI., page 251.) We can, however, trace this 

 path in a simpler way. 



When a body is projected with any velocity, as, for example, 

 when a bullet is fired from a gun, it is acted upon by two forces 

 the original velocity with which it was started, which, as we are 

 not considering the resistance of the air, we may consider to be 

 a uniform force ; and, secondly, the attraction of the earth, which 

 is an accelerating force, causing it to fall 16 feet in the first 

 second, 48 in the next, and so on. Now from a knowledge of 

 these two motions we can easily tell at what point the body will 

 be at any given moment; 

 and by thus finding seve- 

 ral different points in its 

 course we can trace out 

 its path. 



Let the bullet be pro- 

 jected from the point A 

 (Fig. 101), in the direc- 

 tion A B, with any given 

 velocity, and take A c of 

 such a length as to re- 

 present the space it would 

 pass over in one second. 

 Draw A D vertically 

 downwards to represent 

 16 feet, and complete the 

 parallelogram A D E c. 

 AC and AD represent, then, the two forces acting on the bullet; 

 and, since each produces its full effect, it will at the end of one 

 second have arrived at the point E. Since, however, the force of 

 gravity is not uniform, the line A E, which represents its path, will 

 not be straight, but curved upwards, for when a half of A c has 

 been passed over, gravity will only have caused it to move over a 

 quarter of A D. If now we draw through E a straight line E r, 

 parallel to A B and equal to A c, it will represent the motion of 

 the bullet from its original impulse during the next second. To 

 represent gravity we must take E G, three times the length of A D, 

 and thus, by completing the parallelogram, we find that at the 

 end of this second the bullet has arrived at H. In the same way, 

 by making H L equal to five times A D, we find K to be the point 

 at which the bullet will have arrived after three seconds, and in 

 this way we can map out its whole path. 



We see from this the reason why the sights of a rifle are 

 arranged as they are. If the bullet travelled in a perfectly 



Fig. 101. 



straight line, the soldier would aim directly at the point he 

 wished to hit ; but the force of gravity acts on the bullet, and 

 therefore he has to point the rifle at a point as much above it 

 as the bullet will fall in the time it takes to travel the distance. 

 If, for instance, it takes two seconds for the ball to reach the 

 target, he must aim at a point 64 feet above it. To do this 

 would be very inconvenient and uncertain, as he would be 

 unable to tell whether the point he was aiming at was directly 

 over the mark. The sight at the end next the stock is therefore 

 made to adjust to different elevations above the barrel, according 

 to the distance of the object aimed at; and thus, though the 

 rifleman sees the two sights in a straight line with it, the 

 barrel is really pointed considerably upwards, as will be evident 

 to a bystander. 



There is one other fact relating to projectiles, which, though 

 it seems strange, is a necessary result of the second law of 

 motion. 



If a body be projected horizontally, no matter how great its 

 velocity ^be, it will always reach the earth in exactly the same 

 time as if it fell vertically. The speed in falling is not in any 

 way interfered with by the horizontal motion. 



EXAMPLES. 



1. A stone is thrown up with a velocity of 100 feet per second. 

 How high will it rise, and how long will it be before it reaches 

 the ground again ? 



2. A bullet takes 5^ seconds to fall from an elevation to the 

 ground. What is the elevation ? 



3. With what velocity will a stone falling from a height of 

 400 feet strike the ground F 



4. How long will a weight take to fall 784 feet, and how far 

 ll it move in the last second ? 



5. What space will a falling body describe in the 17th second ? 



6. How long will a ball dropped from a height require to 

 attain a velocity of 192, and what space will be described in 

 attaining it P 





LESSONS IN ITALIAN. XV. 



AFTER the following vocabulary and exercise, the pupil may 

 study the uses of the particle a. 



