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to his great discovery of the laws of gravity, and of their application to the motions of the heavenly bodies. I think these acorns very naturally suggest a similar inquiry.
Mr. M. Reminded, by these evidences around us, of the constant operation of the laws of gravity, we will begin our lesson. In our last conversation John gave the example of a stone thrown upward as an illustration of retarded and accelerated motion. Can he now explain the cause of the increase of velocity in descending?
4. John. I have read that every body or mass of matter in the universe attracts every other mass, and that as the bodies approach each other the attraction is increased. I think this increase of attraction between the stone and earth, as they come nearer each other, is the cause of the accelerated motion of the stone in falling.
5. Mr. M. You have mistaken the cause of its accelerated motion; for, though it is true that the force of gravity increases as a body approaches the earth, the difference is so trifling at small distances from the surface as not to be perceptible. When a stone falls from a height, the impulse' which it receives from gravitation in the first instant of its fall would be sufficient to bring it to the ground with a uniform velocity, even if the force of gravity were then taken away; but as the force of gravity is exerted during the next instant also, the stone then receives an additional impulse downward, and so during each succeeding instant, and thus the motion is uniformly accelerated.
6. John. It is perfectly plain that, while the first impulse continues, gravity is constantly acting, and thus the velocity of a falling body is increased. I would like to ask if a stone occupies the same time in going up as in coming down.
7. Mr. M. It does; for in going up its force is constantly diminished by gravity. Can you tell me how many feet a 'body will fall in one second of time?
Ida. It is found by experiment that a body falls sixteen feet during the first second.
Mr. M. How far the next second?
8. Ida. Forty-eight feet, making sixty-four feet in the two seconds.
Ella. It appears to me, then, by the principles just stated, that out of the forty-eight feet which the body falls during the second second, sixteen must be owing to gravity, the same as in the first second, and the remaining thirty-two feet to the velocity which the body had acquired in falling the first sixteen feet. Is not that so?
9. Mr. M. That is the correct explanation. The laws of falling bodies may be shown by triangles, as in this diagram,2 Figure 6. The acceleration of a falling mass may be represented by the divergence of the two sides, a, b, and a, c. If we divide the large triangle into smaller ones by the lines 1, 1, 2, 2, etc., which may represent seconds, the bases of the triangles which we thus make will show the acceleration at any required time, and the areas of the several smaller triangles will represent the space fallen through.
5 10. In such case the area of each smaller triangle must be considered 16, the number of
feet which the body falls by the force of gravity alone; and the base must be called 32, the velocity which a body attains in falling 16 feet by the force of gravity alone. You see, by the figure, that in the first second there is one triangle, or the body falls 16 feet, and has a velocity at the end of the time of 32 feet. In the next second it passes over the space of three triangles, or 48 feet, and has a velocity of 64 feet. In the third second we have five triangles, or 80 feet, and a velocity of 96 feet.
In the same manner the velocity and spaces for any subsequent second in the fall of the body may be shown.
11. Frank. I thought the number of feet described5 during any portion of time was the product of the square of the time in seconds multiplied by 16, but I do not see how that follows from the figure.
12. George. I think I understand it. You have only to count the number of triangles above any line, and you will have the square of the time represented by that line. Thus, above the line 3 there are 9 triangles, and as each one is 16, the space above the line 3 is 9 times 16, or 144, which is the
space a body falls in three seconds, though it only falls 80 feet in the third second. Do you not see 5 triangles between lines 3 and 2?
13. Frank. Is it possible that we can extend that figure as far as we please, and work any problems' of falling bodies by it?
Mr. M. Let us try a problem. How high is a flag-staff, if an arrow, thrown as high as its top, is six seconds in the air?
14. George. It will be as long in going up as in coming down; hence it will be falling three seconds. Three squared is nine, and nine times sixteen are one hundred and fortyfour, the height of the staff.
Mr. M. Correctly solved; but with what velocity was the arrow shot upward?
15. George. I see in the figure three bases of the small triangles, and as each one represents 32 feet, the three will be 96 feet, which the arrow acquired in falling; that must equal the velocity which was destroyed by gravity in its ascent.
16. Mr. M. Do you observe from the figure that when the bases are doubled, the whole number of triangles above such bases is quadrupled ?8 Thus, above the line 3 there are 9 triangles, and above the line 6 there are 4 times 9, or 36. George. Can we not prove from that, that if a person doubles his charge of powder he can shoot four times as far upward?
17. John. Yes; and if I start to run up stairs with double the velocity of another boy, I can go up with one half the muscular exertion.
Frank. How can that be?
John. Do you not see that I can go four times as far by doubling the velocity, but to double the speed I must use twice as much force?
18. Mr. M. I have listened with interest to your discussion, and am pleased to see you so readily apprehend the doctrine of falling bodies; and I recommend you to construct and study such figures as the one I have described to you.
Ella. Do not heavy bodies fall swifter than light ones? 19. Mr. M. Practically they do; but if it were not for the
resistance of the air, all bodies would fall in the same times from the same height. I will show you the "guinea and feather" experiment. You observe that in this tall glass receiver, when it is full of air, the coin falls much more rapidly than the feather. I will now replace them, and exhaust the air from the receiver, and what do you observe as I detach 10 them at the same time?
20. Ella. The feather seems as heavy as the coin, for they fell together.
Mr. M. So great is the resistance of the air when great surface is exposed to it, that people have descended from the height of two miles, by the aid of a kind of stout umbrella, called a parachute,11 and gently touched the ground.
21. Frank. I have just read of a balloon which burst at a great elevation, and the aeronaut12 came down in safety, as the balloon was held within the net-work in the form of a parachute.
22. Mr. M. An ancient poet, Lucretius, knew the resist ance of air when he wrote,
"In water or in air, when weights descend,
The heavier weights more swiftly downward tend;
But if the weights in empty space should fall,
23. You have perhaps heard the old proverb that "a child can throw a feather as far as Hercules," and you can doubtless see the reason in the resistance of the air.
There is another matter for explanation in this lesson, concerning what is called the centre of gravity.
Frank. As gravity is weight, the centre of gravity must be the same as the centre of weight, or the point where a body will balance.
24. Mr. M. It is the point about which all the parts balance each other; and if this point be supported, the whole body will be supported; otherwise the body will fall.
Ella. That must be the reason why the stage was upset the other day. Mr. Jones said it was dangerous to pile so many trunks on the top.
25. Mr. M. If you imagine a line drawn from the centre of gravity of
a body, toward the centre of the earth, that line is called the line of direction. If that line fall within the base or support of a body, the body will stand; but if it does not come with
in the base, the body will fall. The centres of gravity in the figures shown in this cut are where the lines cross each other.
26. John. The figure b has the same base as a, but it can be more easily overturned. If it were about four times its present height, I do not think it could stand at all.
Ida. I have read of a leaning tower at Pisa, in Italy, which appears as if it were just ready to fall, and yet it has stood several hundred years.
27. Mr. M. This wonderful tower is one hundred and ninety feet high, and leans twelve feet; but this is less than half of its diameter; hence the line of direction falls within the base, as you see in the drawing which I show you (Fig. 11).
Fig. 11.-Leaning Tower of Pisa.
John. My father has often told me never to stand up small boat when the waves cause it to rock violently, as it would increase the danger of upsetting. I can now see that