That is, C is equal to the velocity at the beginning of the time t, and C' is equal to space passed over up to the same time. These values of the velocity and space are called, respectively, the initial velocity, and the initial space. Substituting for C and C' these values in (65) and (66), they become,

From these equations, we see that the velocity at any time t, is made up of two parts, the initial velocity, and the velocity generated during the time t; we also see, that the space is made up of three parts, the initial space, the space due to the initial velocity for the time t, and the space due to the action of the incessant force during the same time.

By giving suitable values to v' and s', Equations (67) and (68) may be made to express every phenomenon of varied motion. If we suppose both v' and s' equal to 0, the body will move from a state of rest at the origin of times, and Equations (67) and (68) will become,

From the first of these equations, we see that, in uniformly varied motion, the velocity varies as the time; and, from the second one, we see that the space described varies as the square of the time.

If, in Equation (70), we make t = 1, we have,

That is, when a body moves from a state of rest, under the action of a constant force, the acceleration is equal to twice the space passed over in the first second of time.

If, in the preceding equations, we suppose ƒ to be essentially positive, the motion will be uniformly accelerated; if we suppose it to be negative, the motion' will be uniformly

retarded. In the latter case, Equations (67) and (68)

115. THE FORCE OF GRAVITY is the force exerted by the earth upon all bodies exterior to it, tending to draw them towards it. It is found by observation, that this force is directed towards the centre of the earth, and that its intensity varies inversely, as the square of the distance from the centre.

Since the centre of the earth is so far distant from the surface, the variation in intensity for small elevations above the surface will be inappreciable. Hence, we may regard the force of gravity at any place on the earth's surface, and for small elevations at that place, as constant, in which case, the equations of the preceding article become immediately applicable. The force of gravity acts equally upon all the particles of a body, and were there no resistance offered, it would impart the same velocity, in the same time, to any two bodies whatever. The atmosphere is a cause of resistance, tending to retard the motion of all bodies falling through it; and of two bodies of equal mass, it retards that one the most, which offers the greatest surface to the direction of the motion. In discussing the laws of falling bodies, it will, therefore, be found convenient, in the first place, to regard them as being situated in vacuum, after which, a method will be pointed out, by means of which the velocities can be so diminished, that atmospheric resistance may be neglected.

Let us denote the acceleration due to gravity, at any point on the earth's surface, by g, and the space fallen through in the time t, by h. Then, if the body moves from a state of rest at the origin of times, Equations (69) and (70) will give,

(73.)

(74.)

From these equations, we see that the velocities at two different times are proportional to the times, and the spaces to the squares of the times.

It has been found by experiment that the velocity imparted to a body in one second of time by the action of the force of gravity in the latitude of New York, is about 321 feet. Making g = 321 ft., and giving to t the successive values 1, 2, 3, &c., in Equations (73) and (74), we shall have the results indicated in the following

Solving Equation (74) with respect to t, we have,

That is, the time required for a body to fall through any height is equal to the square root of the quotient obtained by dividing twice the height in feet by 32.

Substituting this value of t in Equation (73), we have,

whence, by solving with reference to v and h respectively,

These equations are of frequent use in dynamical investigations. In them the quantity v is called the velocity due to the height h, and the quantity h, the height due to the velocity v.

If we suppose the body to be projected downwards with a velocity v', the circumstances of motion will be made known by the Equations,

v = v' + gt,

h = v't + 1gt2.

In these equations we have supposed the origin of spaces to be at the point at which the body is projected downwards.

Motion of Bodies projected vertically upwards.

116. Suppose a body to be projected vertically upwards from the origin of spaces with a velocity v', and afterwards to be acted upon by the force of gravity. In this case, the force of gravity acts to retard the motion. Making in (71) and (72), s'o, fg, and sh, they become,

In these equations, h is positive when estimated upwards from the origin of spaces, and consequently negative, when estimated downwards from the same point.

From Equation (77), we see that the velocity diminishes as the time increases. The velocity will be 0, when,

become negative, and the body will retrace its path. Hence, the time required for the body to reach its highest elevation, is equal to the initial velocity divided by the force of gravity.

Eliminating t from Equations (77) and (78), we have,

Making 0, in the last equation, we have,

v =

(79.)

(80.)

Hence, the greatest height to which the body will ascend, is equal to the square of the initial velocity, divided by twice the force of gravity.

This height is that due to the initial velocity (Art. 115).

v'

If, in the same equation, we make t =

+t, we find,

Hence, the velocities at equal times before and after reaching the highest points, are equal.

The difference of signs shows that the body is moving in opposite directions at the times considered.

If we substitute these values of v successively, in Equation (79), we shall, in both cases, find

which shows that the points at which the velocities are equal, both in ascending and descending, are equally distant from the highest point; that is, they are coincident. Hence,