THE POPULAR EDUCATOR 



either accelerated or retarded, and if the gain or loss of velocity 

 in equal times be equal, it is said to be a uniformly accelerated 

 or retarded motion. 



A railway train when first started affords an illustration of 

 accelerated motion. The power of the engine is more than 

 sufficient to overcome friction and the resistance of the air, 

 and therefore the speed increases ; but the resistance increases 

 in a greater ratio, till, after a time, it exactly equals the power 

 of the engine, and then equilibrium ensues, and the train con- 

 tinues in a state of uniform motion. 



The actual measurement of the space passed over in a given 

 time is often a difficult thing, especially as there are always 

 counteracting forces which impede the motion in a greater or 

 less degree. There are, however, various ways in which this 

 may be accomplished, some of which we shall see as we proceed. 



Now there are two modes in which we may regard force ; 

 one is, by considering merely the velocity imparted without any 

 reference to the quantity of matter moved ; force considered 

 thus is called accelerating force. The other mode is by taking 

 into account the quantity of matter moved as well as the 

 velocity, and this is called moving force. These are not two 

 diiferent kinds of force, but merely two ways of regarding the 

 same force. It is clear that a different amount of force is 

 required to impart the same speed to two bodies of diiferent 

 weights. The impulse that would impart a very great velocity 

 to a pistol-bullet may scarcely be able to move a large cannon- 

 ball. The quantity of matter or mass of a body is thus an 

 important element in measuring the force required to produce 

 motion in it. Now we cannot determine exactly what the 

 mass of a body is, as we do not know the ultimate particles 

 of which it consists ; but we can always measure it by the 

 weight of the body, for gravity may be considered to act 

 equally on all particles, and therefore two substances on which 

 it acts equally that is, which have the same weight may be 

 considered to contain the same quantity of matter. Hence, 

 when we want to find the quantity of motion or momentum of 

 any body that is, the force which would be required to gene- 

 rate in it a motion equal to its own, or which it would exert 

 against any obstacle which obstructed it we have to multiply 

 its velocity by its weight. 



This is usually given as a definition : The momentum of any 

 body is its mass multiplied by its velocity. If, for example, a 

 body weighing lOOlbs. be moving with a velocity of 15 feet 

 per second, its momentum is 1,500. 



After thus much by way of definition, we pass on to the 

 laws of motion; but we shall have to return to momentum. 

 The most important principles of motion were drawn up by 

 Newton in the shape of three general laws. These have since 

 been altered in their form, but assert nearly the same facts. 

 The first teaches that every body will continue in its state of 

 rest or uniform motion in a straight line unless acted upon by 

 some external force or forces. This law merely asserts the 

 inertia of matter, that is, its inability of itself to alter or 

 modify in any way any motion which has been imparted to it. 

 We can easily understand that a body at rest will remain so 

 unless some force be applied to it, as we see constant illustra- 

 tions of the fact. It is, indeed, one of the earliest truths which 

 we acquire from observation, but the other part of the law 

 seems more at variance with experience. In fact, almost every 

 motion we observe seems at first sight to point out the inaccuracy 

 of the law; but it is only at first sight, and a little examination 

 will show its truth. Let a stone be rolled along the ground 

 with great speed, it comes to rest in a very short time ; so, too, 

 a boat when rapidly rowed along soon stops if the man ceases 

 to ply the oars. The true reason, however^ why in these and 

 similar instances the motion ceases, is, that other forces 

 neutralise that which has been acquired. In the first case, 

 these forces are friction along the ground and the resistance 

 of the air; in the second, the resistance of the water, for the boat 

 as it advances must displace some of the water, and all the 

 momentum it had acquired is thus soon dispelled. If all such 

 counteracting causes could be removed, the body would move on 

 for ever. This cannot, of course, be proved directly by experi- 

 ment, but we can easily assure ourselves of its truth, for, in 

 proportion as we remove these obstructions, the motion con- 

 tinues for a longer period. If, instead of rolling the stone along 

 the ground, we send it on smooth pavement, the motion will 

 continue to a much greater distance ; and if we try the experi- 



ment on a good surface of ice, it will move farther still, the 

 simple reason being that the force of friction which before 

 overcame its motion has been greatly removed. 



From experiments like these we can ascertain the truth of the 

 law, and it is important to bear it in mind, since the neglect of 

 it has often led to great mistakes. 



Force, then, is not required to maintain motion, but only to 

 produce or alter it, either by increasing or diminishing its speed, 

 or by changing its direction. 



We now turn to the second law of motion, which may be 

 stated thus : When any number of forces act on a particle, 

 each produces its full effect in producing or altering motion, 

 exactly as it would if it acted singly on the body when at rest. 

 Of this we have many simple proofs. Let a stone be dropped 

 from the mast-head of a ship, it will fall exactly at the foot of 

 the mast, just as if the vessel were perfectly at rest. 



If gravity alone acted upon it, it would reach the deck some 

 distance in the rear of the mast, for in the interval which it has 

 occupied in falling, the vessel has been moving onwards, and the 

 point from which the stone fell is, when the stone reaches the 

 deck, vertically over a place some distance behind the mast ; but 

 another force was also acting on the stone, and that was the 

 onward motion which, like the vessel, it had acquired. This 

 motion was exactly equal to that of the vessel, as both were 

 moving at the same rate ; and each of these forces produces its 

 full effect. The stone falls in exactly the same time as it would 

 take if the vessel were at rest; it moves through the same 

 horizontal space that it would if it were not falling ; and at the 

 end of the time occupied in falling is in the same place as if 

 each force had acted singly during that length of time, the only 

 difference being that then it would have passed over two sides of 

 a parallelogram, whereas now it has travelled down the diagonal. 



Another good illustration of this is afforded by a boat cross- 

 ing a river when the stream is running down rapidly. Suppose 

 the stream to be flowing in the direc- B c, 



tion of the arrow. A boatman at A "=j 

 wants to cross to a point B some dis- \ 



tance lower down ; he does not, how- ^E 

 ever, steer directly for it, since, if he ^f 

 did, the force of the stream would carry OE^; j 

 him to some point much lower down, 

 but he makes for a point almost oppo- 

 site him. If the current be so rapid that it would carry him 

 down from C to B in the time it takes him to row from A to C, he 

 must steer directly across to c. There will be then two forces 

 acting on the boat his own force impelling it from A to C, and 

 the force of the stream from c to B, and under the joint action 

 of these two forces it will move from A to B in the same time 

 that it would take him to row to c. If, now, he wants to cross 

 again to D he must steer for some point higher up than A, for 

 as B A is longer than A c, the tide will have more time to act 

 upon the boat and carry it down. More commonly, however, he 

 rows from B towards C along the shore, where the current has 

 less force, and then crosses as at first. But it is clear that in 

 iither case each force produces its full effect. 



In our lessons on statics we learnt the parallelogram of forces, 

 and found that if two forces acting on a body be represented 

 by two adjacent sides of a parallelogram, the resultant will be 

 represented by the diagonal. We may now extend this principle 

 to velocities, thus : If any two velocities impressed on a particlo 

 be represented by two sides of a parallelogram, the diagonal 

 will represent the resulting motion in direction and velocity. 



We now pass on to the third law of motion, which was stated 

 by Newton as follows : Reaction is always equal and contrary 

 to action, or the mutual actions of two bodies upon each other 

 are always equal and in opposite directions. When a carriage 



drawn by horses, they are pulled back with the same force as 

 the carriage is drawn forward ; so, if a boat in a stream be 

 pushed off from another, the quantity of motion produced in 

 iach is the same. If both be of the same weight they will move 

 with the same velocity ; but if one be heavier, its motion will be 

 so much less than that of the other. We see, thus, that motion 

 is never lost, it always produces motion in other things ; but as 

 this is shared among all bodies in proportion to their mass, it 

 soon becomes so small as to be unnoticed. 



Now if we consider the pressure on a body to be the action, 

 bhe quantity of motion produced is the reaction, and this law 

 asserts that these are equal. But the quantity of motion is 



Fig. 05. 



