may be regarded as a negative acceleration. A body that starts with a velocity of 100 feet per second, and that has a negative acceleration of 15 feet per second each second, will have a velocity of 100 (3 × 15) = 55 feet per second at the

end of three seconds.

29. The Acceleration produced by Gravity.-Experiment shows that a heavy body, when allowed to fall freely, falls vertically

so that, if we multiply the square of the number of seconds by 16, we get the distance in feet through which a falling body descends in a given time from rest.

Now, since gravity is a uniform continuous force acting upon a body, it must act as a uniform accelerating force on a falling body. And since experiment shows a body falls through 16 feet in the first second when it starts from rest, its velocity at the end of the first second must be as much faster than 16 feet as it was slower at the beginning. At the beginning the velocity was zero, so that to pass from rest through 16 feet in one second with the velocity uniformly increasing, the velocity at the end of the first second must be 32 feet per second. Gravitation near the earth's surface, being continuous and uniform, will therefore cause a velocity of 32 feet per second during each second that it acts—that is, the acceleration produced by gravitation is 32 feet per second per second. The number denoting the acceleration due to gravity is often denoted by the letter g.

Remembering the distance fallen through in one second, and the velocity acquired at the end of the first second, a little thought will enable the reader to find the distance traversed in any succeeding second, or the velocity acquired at the end of any number of seconds. Thus at the end of the first second a falling body has a velocity of 32 feet, and if gravity ceased to act it would pass through this distance in

the second second. But gravity continues to act, and makes it fall through 16 feet in addition to the 32 feet-through 32+16 48 feet in the second second.

=

Add this to the 16 feet fallen through in the first second, we get 48 + 16 = 64 feet, the distance fallen through in two seconds. In the second second gravity adds another 32 feet of velocity, so that the velocity at the end of the second second is 32 × 2 = 64 feet per second.

On throwing a ball or other object vertically upwards, gravity acts as a uniformly retarding force until it comes to rest, when gravity brings it back with uniform acceleration. A little reflection will show that a body thrown upwards takes as long to fall as it does to rise, that the height which it reaches is just equal to the distance it would fall from rest during the time it occupies in rising, and that its initial upward velocity will be the same as its final downward velocity.

30. Inertia and the First Law of Motion.-Observation and experiment show that matter has no power of itself to change its own state of rest or of motion; in other words, that force is required to move any piece of matter or to alter its rectilinear motion in any way. This implies that when a body is at rest, a certain resistance must be overcome by the application of an external force before the body can be set in motion, and that if a body is already in motion and we wish to make it move faster or slower or to change the direction of its motion, a force must still be applied, because in each case resistance is to be overcome. The property possessed by all matter of offering resistance to change of state, either of rest or motion, is called inertia. This amounts to saying—

Inertia is that property of matter which makes the application of external force necessary before any motion or change of motion can be produced in a body.

A description and account of the property of inertia is contained in the first of the three laws of motion as given by Newton. The first law of motion states: Every body continues in its state of rest or of uniform motion in a straight line, except in so far as it is compelled to change its state by external forces acting upon it.

The fact that a body at rest remains at rest when not acted upon by an external force is plainly true and a matter of daily experience. A stone or leaf at rest never begins to move without being acted upon by a force. But the law also states

that a moving body would continue to move for ever in a straight line with uniform speed, provided no external force acts upon the body. What is the evidence for the truth of this part of the law? We cannot prove it by direct experiment, as every moving body we are acquainted with is subject to the action of several forces-gravity, friction, the resistance of the air, etc. But the nearer we approach to the conditions under which the second part of the law is stated—a body moving and not acted upon by any force-the more nearly does the body continue to move in a straight line with uniform speed. A smooth ivory ball thrown along a rough horizontal grass plane is soon brought to rest by the opposing forces, the friction of the plane and the resistance of the air. Throw it along a large horizontal plane of wood with the same force, and it travels farther, as the friction is then less. Throw it along a smooth sheet of ice, and it remains in motion still longer. Hence we conclude that if we could have a perfectly smooth horizontal surface in a space devoid of air, the ball would move at a uniform speed in a straight line for ever, for a body has no power in itself to alter either its speed or its direction.

Many illustrations of the first law of motion and of the property of inertia are met with in ordinary life. When a person is standing up in a carriage at rest, and the carriage is suddenly moved forwards, the person falls backwards, because, not being fixed to the carriage, his inertia tendency to keep his state causes him to remain at rest in the position he occupies. This is an example of the tendency of a body to remain at rest when not acted upon by force"inertia of rest," as it may be called.

or

As an example of the "inertia of motion," or the tendency of a moving body to continue moving uniformly in a straight line when not acted on by a force, consider first a rider upon a moving object (horse or carriage) that suddenly stops. His inertia or tendency to keep his state of motion causes him to

move on so that he falls forward, for the force which stops the carriage is not applied to the rider to the same extent. If a carriage make a sudden turn when moving rapidly, the rider is liable to be thrown on the outer side or the carriage overturned, owing to the tendency of a moving body to continue its motion in a straight line.

31. Force.-Newton's first law of motion implies the following definition of force :

Force is any cause which changes or tends to change a body's state of rest or of uniform rectilinear motion.

The definition of force does not tell us what forces are in themselves, but merely what effects are produced by the causes called forces. It must be noticed that a force does not always produce motion, for if counteracted by another force it will only "tend" to change a body's state of rest or motion. Forces that can set bodies in motion, such as attractions, the push or pull of a living agent, or a compressed spring may be called active forces; forces that are only able to check or prevent motion, such as friction and all other kinds of resistance, may be called passive force.

Forces may be measured in various ways. One way of measuring a force is to find the velocity it can impart to unit mass in a unit of time, that is, to find the acceleration it can produce. Another mode of measuring a force is to compare it with a unit weight, so that a force is often described as a pressure of so many pounds' weight.

Forces may often be conveniently represented by straight lines, for a straight line of finite length can represent (1) the point of application of a force, (2) its direction, and (3) its magnitude. For example

(1) The point A represents the point of application of a force F.

(2) The direction

from A to B represents

FIG. 17.

the line of action and direction of the force.

F

(3) The length AB represents the magnitude of the force when it contains as many units of length as the force contains units of force.

Any unit of length, as an inch or

inch, may be chosen in

any particular case to represent a unit of force, and then a line of twice that length will represent a force twice as great, and so on. 32. Newton's Second Law of Motion. The second law of motion may be stated thus: The rate of change of momentum of a body is proportional to the external force producing it, and takes place in the same direction as the force acts.

A "rate of change" of any quantity is measured by the amount of change that takes place in a unit of time.

Momentum, or Quantity of Motion, denotes the product of the mass of a moving body into the velocity with which it moves. We may take as the unit of momentum a body of I lb. which is moving with a uniform velocity of 1 foot in a second. Then the numerical value of the momentum of a body is the product of the number of pounds in it by the number of units of velocity with which it is moving. Experiment shows that when force produces motion in a body, the momentum produced in one second is proportional to the force. Hence force can be measured by the momentum it is capable of producing in a unit of time. A ball of lead weighing 10 lbs. and moving with a velocity of 18 feet a second would strike an obstacle with the same force as a ball weighing 30 lbs. and moving with a velocity of 6 feet a second. The momentum or mass-velocity in both cases equals 180.

In general, momentum = mass × velocity, and as momenta are exactly proportional to the forces producing them, we can compare forces by comparing the momenta produced by them.

The second law implies that each force acting on a body produces its own change of momentum of the body, independently of any other force or forces that may be acting. This leads us to the consideration of the composition of two forces, or the reduction of two forces to one resultant.

33. Composition of Forces.-We often wish to know what effect two or more forces have upon a body, and thus to find out what single force will produce the same result as all the others combined. This single force is called the resultant, and the problem of finding it is known as the Composition of Forces.

(1) Suppose two forces to act in the same direction upon a