Thus, referring back to the summary preceding [Lecture IV], it is there stated that a dropped body falls 16 feet in the first second, that in two seconds it falls 64 feet, and so on, in proportion to the square of the time. So also will it be the case with a thrown body, but the drop must be reckoned from its line of motion—the straight line which, but for gravity, it would describe.
Thus a stone thrown from O with the velocity OA would in one second find itself at A, in two seconds at B, in three seconds at C, and so on, in accordance with the first law of motion, if no force acted. But if gravity acts it will have fallen 16 feet by the time it would have got to A, and so will find itself at P. In two seconds it will be at Q, having fallen a vertical height of 64 feet; in three seconds it will be at R, 144 feet below C; and so on. Its actual path will be a curve, which in this case is a parabola. ([Fig. 57.])
If a cannon is pointed horizontally over a level plain, the cannon ball will be just as much affected by gravity as if it were dropped, and so will strike the plain at the same instant as another which was simply dropped where it started. One ball may have gone a mile and the other only dropped a hundred feet or so, but the time needed by both for the vertical drop will be the same. The horizontal motion of one is an extra, and is due to the powder.
As a matter of fact the path of a projectile in vacuo is only approximately a parabola. It is instructive to remember that it is really an ellipse with one focus very distant, but not at infinity. One of its foci is the centre of the earth. A projectile is really a minute satellite of the earth's, and in vacuo it accurately obeys all Kepler's laws. It happens not to be able to complete its orbit, because it was started inconveniently close to the earth, whose bulk gets in its way; but in that respect the earth is to be reckoned as a gratuitous obstruction, like a target, but a target that differs from most targets in being hard to miss.
Fig. 58.
Now consider circular motion in the same way, say a ball whirled round by a string. ([Fig. 58.])
Attending to the body at O, it is for an instant moving towards A, and if no force acted it would get to A in a time which for brevity we may call a second. But a force, the pull of the string, is continually drawing it towards S, and so it really finds itself at P, having described the circular arc OP, which may be considered to be compounded of, and analyzable into the rectilinear motion OA and the drop AP. At P it is for an instant moving towards B, and the same process therefore carries it to Q; in the third second it gets to R; and so on: always falling, so to speak, from its natural rectilinear path, towards the centre, but never getting any nearer to the centre.
The force with which it has thus to be constantly pulled in towards the centre, or, which is the same thing, the force with which it is tugging at whatever constraint it is that holds it in, is mv2/r; where m is the mass of the particle, v its velocity, and r the radius of its circle of movement. This is the formula first given by Huyghens for centrifugal force.