is due to the velocity with which the body commenced the third second.
518. We see therefore that after the lapse of two seconds gravity has communicated to the body a velocity of 64' per second; we should similarly find, that at the end of the third second, the body has a velocity of 96', and in general at the end of t seconds a velocity of 32t. Thus we illustrate the remarkable law that the velocity developed by gravity is proportional to the time.
519. This law points out that the most suitable way of measuring gravity is by the velocity acquired by a falling body at the end of one second. Hence we are accustomed to say that g (as gravity is generally designated) is 32. We shall afterwards show in the lecture on the pendulum (XVIII.) how the value of g can be obtained accurately. From the two equations, v = 32t and s = 16t² it is easy to infer another very well known formula, namely, v² = 64s.
THE PATH OF A PROJECTILE IS A PARABOLA.
520. We have already seen, in the experiments of [Fig. 68], that a body projected horizontally describes a curved path on its way to the ground, and we have to determine the geometrical nature of the curve. As the movement is rapid, it is impossible to follow the projectile with the eye so as to ascertain the shape of its path with accuracy; we must therefore adopt a special contrivance, such as that represented in [Fig. 70].
b c is a quadrant of wood 2" thick; it contains a groove, along which the ball b will run when released. A series of cardboard hoops are properly placed on a black board, and the ball, when it leaves the quadrant, will pass through all these hoops without touching any, and finally fall into a basket placed to receive it. The quadrant must be secured firmly, and the ball must always start from precisely the same place. The hoops are easily adjusted by trial. Letting the ball run down the quadrant two or three times, we can see how to place the first hoop in its right position, and secure it by drawing pins; then by a few more trials the next hoop is to be adjusted, and so on for the whole eight.
521. The curved line from the bottom of the quadrant, which passes through the centres of the hoops, is the path in which the ball moves; this curve is a parabola, of which f is the focus and the line a a the directrix.
Fig. 70.
It is a property of the parabola that the distance of any point on the curve from the focus is equal to its perpendicular distance from the directrix. This is shown in the figure. For example, the dotted line f d, drawn from f to the centre of the lowest hoop d, is equal in length to the perpendicular d p let fall from d on the directrix a a.