subject to the uniform acceleration f,—and R being caught off at S, P will move on through ST uniformly with the velocity acquired. The times occupied in moving through AS and ST are observed with considerable accuracy by a contrivance of clock-work attached to the machine.

83. The results of numerous experiments made with Atwood's machine, lead to the conclusions that gravity has a uniform accelerating effect, and that its numerical value is that stated in Art. 43. The most trustworthy results however are (as there stated) to be obtained from experiments on pendulums, but they are of too refined a character to be discussed here.

CHAPTER IV.

OF THE MOTION OF PROJECTILES.

84. In the present Chapter we shall consider:

(i) The projectile as a single heavy particle, (ii) that the accelerating force of gravity is uniform, and acts in the same direction at all points of the path of the projectile; (iii) that the effect of the rotation of the earth is neglected, and (iv) that the motion takes place in vacuo-no account being taken of the resistance of the air. See Art. 94.

85. A body projected in any direction which is not vertical, and acted on by the force of gravity only, will describe a parabola.

T

Let the body be projected from the point A in direction AT with velocity v. Draw AV vertical and downwards, and let P be the position of the body at any time t after the instant of projection.

Let the motion of the body be referred to the directions AT, AV (Art. 19), and draw PT,

A

H

N

PV parallel to AV, AT;-now the motion being at any and every instant referred to the directions AT, AV-the force of gravity will have a uniform accelerating effect (g) in direction AV and there will be no acceleration in direction AT, we shall have therefore

This relation between PV and AV shews that the path AP is a parabola whose axis is vertical, and directrix consequently horizontal; AV being a diameter, and AT the

tangent at A, the parameter at A being

=

2v2

COR. 1. If h be the space due to the velocity of projection v, (i. e. the space through which a body must fall freely from rest under the action of gravity, in order to acquire the velocity v,) v2 = 2gh; wherefore PV24h. AV. Hence 4h is the parameter at A, and therefore h is equal to the vertical distance of A below the directrix.

COR. 2. The result of Cor. 1 may be thus interpreted: "The velocity of projection of a projectile is the same as would be acquired by a body falling freely from the directrix to the point of projection.'

And further, since the body after passing through any point of its path will move in the same way as if it had been projected from that point with the velocity it then has, and in the direction in which it is then moving,-hence, "the velocity of a projectile at any point P of its path is equal to that due under the action of gravity to the vertical distance of that point from the directrix."

Obs. Let the horizontal plane which passes through the point of projection A meet the parabola again in H, and let T be the time of passing from A to H, AH = R,—then

R, T are called the range, and time of flight of the projectile on the horizontal plane; and if a be the angle which the direction of projection makes with the horizon, the angle a is called the elevation of the projectile.

T

86. If the motion of the body be estimated vertically and horizontally—along Ay and Ax,—the velocity of projection vertically is usina, and horizontally is v cos a;-the horizontal velocity will remain uniformly equal to v cos a during the motion, since there is no force in direction Ax; the vertical velocity will gradually be reduced to zero by the action of gravity, and the body is then

A

H

at its greatest height z above the horizontal plane AH, but the continued action of gravity will generate velocity downwards, and bring the body to the plane at H after a time equal to that in which it moved from A to the highest point. We shall have the following results,

if be the time of moving from A to the highest point

T

= time in which the initial vertical velocity v sin a is destroyed by force of gravity g,

π

This result is the same if -a be put for a; shewing

2

that there are two directions in which a body may be projected with a given velocity, so as to have the same horizontal range.

For a given velocity of projection v, the horizontal range R will be greatest when sin 2a = 1; i. e. when a = 45°.

Again, the latus rectum of the parabola is the parameter at the highest point, and the velocity at the highest point being =v cosa, the distance of that point from the directrix is

87. To find the range (R) and time of flight (T) of a pro

and these two resolved parts of gravity are constant.

N

Hence if T be twice the time in which the velocity