NATURAL PHILOSOPHY MATTER, MOTION, AND HEAT. 



spaces, or 80 feet ; and that the whole distance 

 fallen during all the three seconds is 9 spaces, or 

 144 feet. 



The distances that bodies fall, then, do not 

 increase simply as the times, but as the squares of 

 the times. To find, therefore, how far a body will 

 fall in any number of seconds, multiply the num- 

 ber of seconds by itself, and that product by 16. 

 Thus, in 7 seconds, a body will fall 7 x 7 x 16 = 

 784 feet. The height of a precipice might be 

 roughly measured in this way, by observing how 

 many seconds a stone takes to reach the bottom. 



As a body in descending to the earth receives 

 increasing accessions to its velocity during every 

 successive second, so, when a body is projected 

 upwards from the surface of the earth, its velocity 

 decreases in the same proportion till it comes to a 

 state of momentary rest, when it instantly begins 

 to descend with a gradually increasing velocity, 

 which at any point in the descent is equal to its 

 velocity at the same point when ascending. In 

 this calculation, however, we omit the influence of 

 the atmosphere. 



Projectiles. 



Any heavy body launched or projected into 

 the air with an impulse, is a projectile ; and to 

 investigate the motions of such bodies forms a 

 distinct branch of the science of motion. 



Suppose a cannon-ball fired from the top of a 

 tower, whose height is represented by the line A4', 

 in the direction of the horizontal line AB. The 



B 



Fig. 12. 



force of the gunpowder ceases to act as the ball 

 leaves the cannon, and the ball retains a uniform 

 velocity which would carry it over equal spaces in 

 equal times. Let Ai be the space the ball would 

 describe in a second if acted on only by the pro- 

 jectile force, and set off a number of equal spaces 

 along AB. Now, from the moment the ball leaves 

 the mouth of the cannon, it is acted on by gravity ; 

 and while moving in the direction of AB, also 

 falls ; so that at the end of a second it must be 16 

 feet lower than it would have been if gravity had 

 not acted that is, if we take the perpendicular 

 line if to represent 16 feet, the ball will at the 

 end of the first second be at c. The horizontal 

 motion being uniform, the ball at the end of the 

 second second would be at 2, or twice as far 

 towards B ; but its downward accelerated motion 

 has in the meantime dragged it down through 

 three spaces more in all four spaces of 16 feet 

 below the horizontal line. Making 26, therefore, 

 equal to four times ic, gives b as the point where 

 the ball will be at the end of the second second. 

 By similar reasoning, if 3^ is made equal to nine 

 times ic, 4f equal to 16 times ic, &c. according to 

 the law of accelerated motion ; <?, /, &c. will be 

 the points at which the ball will be found at the 



end of the third, fourth, &c. seconds. The result- 

 ant of these two dissimilar motions, the one uni- 

 form, and the other uniformly accelerated, is a 

 curve of the kind called a parabola, as represented 

 in the figure. 



From the above investigation, this remarkable 

 consequence follows : that a body projected hori- 

 zontally from a height above a level plane, comes 

 to the ground as soon as if it were let fall perpen- 

 dicularly. The ball reaches / in the same time 

 that it would fall from A to 4'. A greater velocity 

 of projection would make it take a wider flight ; 

 but its horizontal motion does not interfere with 

 its downward motion, and at the end of four 

 seconds it must be at some point in the same 

 horizontal line at g, for example. Two balls, 

 then, projected horizontally from the same height 

 above a level plane, though the one may range 

 only a mile, and the other two miles, will reach 

 the ground at the same time. 



Projectiles are mostly thrown, not horizontally, 

 but in an oblique direction. We arrive, how- 

 ever, at the path described in the same way as in 

 the last case. 



The laws above arrived at respecting projec- 

 tiles are strictly true only on the supposition that 

 the movements are made in empty space. But 

 every projectile has to encounter the resistance 

 of the air ; and that resistance becomes so great 

 when the velocity is very high, that in the practice 

 of gunnery the theory is of little value. With 

 a small velocity, however, a body thrown through 

 the air describes a path not differing much from a 

 parabola. 



Since the distance of the flight, and the width 

 of the curve described by a projectile, increase 

 with its initial velocity, we can conceive the 

 velocity increased until the curve became as large 

 as that of the earth itself. If this were the case, 

 the projectile, instead of falling, would, if the 

 resistance of the air were removed, continue to go 

 round and round the earth for ever. We thus 

 arrive at the idea of planetary motion. The moon, 

 for instance, is constantly falling towards the 

 earth, like a cannon-ball shot horizontally ; but the 

 projectile velocity which it had from the begin- 

 ning, and which, as it moves in empty space, it 

 retains undiminished, is sufficient to carry it clear 

 round in the curve called its orbit. 



Centre of Gravity. 



The centre of gravity of a body is that point 

 about which the body balances itself in all posi- 

 tions. When a rod of uniform thickness and 

 density is suspended by its middle point, it 

 remains at rest. There is as much matter on the 

 one side as the other ; the atoms balance one 

 another in pairs ; and it is as if the whole were 

 collected at the point of suspension. The centre 

 of gravity of such a rod is the central point at its 

 middle part ; and if the substance of the rod is 

 pierced, and this central point rested on a fine 

 needle, the rod will remain immovable in what- 

 ever position it is put. 



Whenever bodies are regularly shaped, and of 

 uniform density, their centres of gravity may be 

 found by mathematical measurement, as in the 

 case of the rod. Thus the centre of gravity of a 

 globe is evidently in its middle point, or centre 

 of dimension. 



