fractional part of the whole weight of the body as the height of the inclined plane is of its length. The acceleration is therefore constant.

h

Since F= Ma, we have a = 9 1

or a=g sin c.

To find the velocity which the body acquires in sliding the length of the plane l, we have only to use the formula (4) of $96. 2as = v2- u2

The body starts from rest, hence u=o and s=l in this case, therefore

but this is precisely the velocity which a freely falling body will gain in falling through a vertical distance h, and there is nothing

in the result which depends on the slope of the plane, therefore the velocity gained by a body in sliding down a frictionless inclined plane of any slope whatever is the same as that gained by a body in falling freely the same vertical distance.

Since the velocity does not depend on the slope of the plane, it will be the same at B (Fig. 49) for any smooth, frictionless curve down which it may slide from A, and it will be the same at B as at C or D.

The time of descent, however, from A to B depends on the curve and may be proved to be a minimum when A and B are joined by the arc of a cycloid.

104. Unit of Work or Energy.-The unit of work on the C. G. S. system of units where the force is measured in dynes and the distance in centimeters is known as the erg (from the Greek word for work). It is the work done when a body moves one centimeter in the direction in which it is urged by a force of

one dyne. Thus about 980 ergs of work are spent in raising a gram weight one centimeter.

The corresponding unit of work or energy on the foot-poundsecond system is the foot-poundal, and is the work done when a body moves one foot in the direction in which it is urged by a force of one poundal.

105. Kinetic Energy. We will now calculate the effect of a certain amount of work in giving motion to a mass m. Suppose a force of F dynes acts on m in the direction of its motion while it is moving through a space of s centimeters; the work done is by definition, F's dyne-centimeters or ergs. But while the constant force Facts there is a constant acceleration a and the equations of $96 therefore apply to the motion, and we have

The change from mu2 to mv2 therefore expresses the amount of work required to change the velocity of the mass m from u to v. Starting from rest, the energy required to give it a velocity v is mv2; this is also the measure of the work that the body can do before coming to rest again, therefore the quantity mv2 is the measure of the kinetic energy or energy of motion possessed by a mass m moving with velocity v.

106. Velocity at Foot of Inclined Plane. The principle of the conservation of energy may be applied to motion on an inclined plane and leads at once to the conclusion previously stated, $103, that the velocity of a body at the foot of an inclined plane depends only on its height and is independent of the slope.

For the work done in lifting the body from the bottom of the plane to the top depends only on the height of the plane, since the work is done only against gravity and serves to increase the potential energy of the body. In sliding down the plane, if no

work is done against friction, all the potential energy gained will be transformed into kinetic energy, so that when it reaches the bottom its kinetic energy will be equal to the work that was done in lifting it. The kinetic energy, of the body and consequently its velocity will therefore be independent of the slope of the plane.

The work done in lifting the mass m the height of the plane h is mgh, for mg is the weight of the mass expressed in dynes. The kinetic energy of the mass at the bottom is mv2, hence mghmv and v2=2gh.

107. Kinetic Energy and Momentum Compared.-Kinetic energy and momentum are both quantities that depend on the mass and velocity of the moving body, but while kinetic energy is expressed by m2 and measures the work done on the body in giving it motion, momentum, expressed by mv, measures the impulse given to it, or the product of the force by the time during which it was acting on the body, for the second law of motion (§42) gives the relation

Ft-mv-mu,

or change in momentum is equal to the impulse when the force is measured in the appropriate unit.

Hence if a force acts upon a body through a certain distance and it is required to find the change in velocity of the body, the formula for kinetic energy must be used,

Fs=mv2-mu2;

while if the time of action of the force is given, the change in velocity is found from the equation of momentum,

Ft-mv-mu.

108. Impact. When one freely moving body strikes against another there is said to be impact. The laws of impact may be studied by the apparatus shown in figure 50, in which two balls are hung as pendulums so that when one is allowed to swing it will strike against the other, and the arcs through which they move are observed on a graduated scale. It will be shown later that when two pendulums are of the same length their velocities at the lowest point are very nearly proportional to the lengths of the arcs through which they swing, their velocities at the instant of impact can thus be readily compared.

When the ball B is at rest and A is allowed to swing against it, if the bodies are inelastic like two balls of lead or putty, they will keep together after impact, the forward momentum of the combined mass being equal to the momentum of A before impact. If the two balls are perfectly elastic or resilient and of equal masses, like two ivory billiard balls,

A will come to rest giving up its whole momentum to B, which will therefore swing out just as far as A has fallen. If the masses are elastic but not equal, then A may continue forward. or have its motion reversed at the instant of impact depending on whether B is the less or greater mass. In all cases of impact, whether the masses are elastic or inelastic, the total momentum of the two bodies is not changed by the impact. That is, if one body loses forward momentum the other gains an exactly equal forward momentum. Stated algebraically,

FIG. 50.

where A and B are the two masses, respectively, while v and u are their velocities before impact, and V and U are their velocities after impact.

This law is easily seen to be a direct consequence of the laws of motion. For at each instant during impact the forward pressure of A upon B is equal to the backward pressure of B against A, as expressed in the statement that action and reaction are equal and opposite. Hence the total forward impulse given to B is equal to the backward impulse sustained by A, and by Newton's second law the change in the momentum of A must be equal and opposite to the change in the momentum of B, consequently the sum of the momenta of the two is not changed. If the two are inelastic they move together after impact with a common velocity x, whence

In case of elastic bodies there is a certain instant during the impact when the compression is a maximum and the two bodies

B

are neither approaching nor receding from each other. At that instant they are moving with the same velocity x which they would have acquired if quite inelastic. But suppose they are perfectly resilient and the pressure between them at any instant as they spring apart is exactly equal to what it was during the corresponding instant of compression. The total backward impulse given to A will then be twice what it would have been if the bodies had been inelastic, hence the total change in velocity of A will be twice as great

FIG. 50a.

as v-x, or 2(v-x), and its final velocity V will be

V=v-2(v-x)=v+2(x-v),

so also

U=u+2(x-u).

If the above expressions are written

the coefficient

V=v+μ(x − v)
U=u+μ(x-u)

will serve to indicate the degree of resiliency. If μ= 1 the bodies are quite inelastic for V = x and U=x, but if μ=2, the resiliency is perfect.

109. Series of Impact Balls.-In case of a row of equal elastic balls the impulse is transmitted from one to another so that if one ball strikes one end of the series, the last ball will fly off as if it had been struck directly. If two balls moving together hit one end of the series, they will come to rest and the last two will fly off. If three moving together strike one end, three will fly off from the other end, etc.

Of course the momentum after impact must be the same as before in every case, but if two balls moving with velocity v struck one end of the series the momentum would not be changed if the last ball alone were to fly off with velocity 2v. But in that case the kinetic energy after impact would be double that before impact, which is of course contrary to the principle of the conservation of energy. If the energy after impact is to be the same as before, the same number of balls must fly off from one end as hit the other.