CONVERSATION VI.
OF THE ATTRACTION OF GRAVITATION.
E. The term momentum, which you made use of yesterday, is another word which I do not understand.
F. If you have understood what I have said respecting the velocity of moving bodies, you will easily comprehend what is meant by the word momentum.
The momentum, or moving force of a body, is its weight multiplied by its velocity. You may, for instance, place this pound weight upon a china-plate without any danger of breaking it, but, if you let it fall from the height of only a few inches, it will dash the china to pieces. In the first case, the plate has only the pound of weight to sustain ; in the other, the weight must be multiplied with the velocity, or, to speak in a popular manner, into the distance of the height from which it fell.
If a ball a lean against the obstacle b, it will not be able to overturn it, but if it be taken up to c, and suffered to roll down the inclined plane d e against b, it will certainly overthrow it ; in the former case, b would only have to resist the weight of the ball a, in the latter it has to resist the weight multiplied into its motion, or velocity.
C. Then the momentum of a small body, whose velocity is very great, may be equal to that of a very large body with a slow velocity.
F. It may, and hence you see the reason why immense battering-rams, used by the ancients in the art of war, have given place to cannon-balls of but a few pounds weight.
C. I do, for what is wanting in weight, is made up by velocity.
F. Can you tell me what the velocity a cannon-ball of twenty-eight pounds must have, to effect the same purposes as would be produced by a battering-ram of 15,000 pounds weight, and which, by manual strength, could be moved at a rate of only two feet in a second of time ?
C. I think I can :- the momentum of the battering-ram must be estimated by its weight, which is 15,000 multiplied by two feet, equal to 30,000 ; now if this momentum, which must also be that of the cannon-ball, be divided by the weight of the ball, it will give the velocity required ; and 30,000 divided by twenty-eight, will give for the quotient 1072 nearly, which is the number of feet which the cannon-ball must pass over in a second, in order that the momenta of the battering-ram and the ball may be equal, or, in other words, that they may have the same effect in beating down an enemy's wall.
E. I now fully comprehend what the momentum of a body is, for if I let a common trap-ball accidentally fall from my hand upon my foot, it occasions more pain than the mere pressure of a weight several times heavier than the ball.
F. If you let a pound, or 100 pounds, fall on the floor, only from the height of an inch and a quarter, it will strike the floor with a momentum equal to double its weight ; and if you let it fall from four times that height, or five inches, it will have double that effect ; and if it fall nine times that height, or eleven inches and a quarter, it will have treble the effect ; and by falling sixteen times the height, or twenty inches, it will have eight times more effect in causing pain than the mere pressure of the ball itself.
C. If the attraction of gravitation be a power by which bodies in general tend towards each other, why do all bodies tend to the earth as a centre ?
F. I have already told you that by the great law of gravitation, the attraction of all bodies is in proportion to the quantity of matter which they contain. Now the earth, being so immensely large in comparison of all other substances in the vicinity, destroys the effect of this attraction between smaller bodies, by bringing them all to itself. If two balls are let fall from a high tower at a small distance apart, though they have an attraction for one another, yet it will be as nothing when compared with the attraction by which they are both impelled to the earth, and consequently the tendency which they mutually have of approaching one another will not be perceived in the fall. If, however, any two bodies were placed in free space, and out of the sphere of the earth's attraction, they would in that case assuredly fall toward each other, and that with increased velocity as they came nearer. If the bodies were equal, they would meet in the middle point between the two; but if they were unequal, they would then meet as much reamer the larger one, as that contained a greater quantity of matter than the other.
C. According to this, the earth ought to move towards falling bodies, as well as they move to it.
F. It ought, and, in just theory, it does ; but when you calculate how many millions of times larger the earth is that anything belonging to it ; and if you reckon the small distances from which bodies can fall, you will then know that the point where the falling bodies and the earth will meet, is removed only to an indefinitely small distance from its surface ; a distance much too small to be conceived by the human imagination.
We will resume the subject of gravity to-morrow.
Reverend Joyce correctly states that if you multiply the mass of an object by its velocity then you'll get its momentum, which is measured in kilogram metres per second or equivalently, Newton seconds. Momentum is a vector quantity (i.e. it has a direction and magnitude, see my last post for that and associated pedantry) which is related to the kinetic energy possessed by a body in motion. Kinetic energy is simply how much energy an object has due to its motion, it is equal to half of the object's mass multiplied by the square of its velocity.
As originally explained by Newton in his second law, momentum is always conserved. In other words, the sum of the momentum of two objects before and after a collision will be identical. One of the simplest ways to demonstrate this is with a Newton's cradle, which is a very simple system that uses pendulums to constrain collisions between identical balls to be in a single straight line. An ideal elastic collision is one in which all momentum (and therefore also all kinetic energy) is conserved. A steel ball striking another steel ball results in a very elastic collision, whereas as collision between a steel ball and a cushion is incredibly inelastic. In the real world, things are very rarely perfect or ideal; some of the kinetic energy is often converted to create sound, heat or to cause the objects to deform in the collision. Crumple zones in a car are a very practical application of removing kinetic energy from a collision, an application which has saved countless lives over the past half century.
When dealing with collisions, it is incredibly useful to consider the change of momentum; which is known as an impulse. An impulse represents the action of a force over time. In collisions, the time period over which a force applies is normally quite short, which can lead to a very high change in momentum being caused by a relatively low applied force.
I've rambled on about collisions mainly because it's core to understanding momentum and how it explains things moving in the world around us. Partly though, it's to be able to clear up what I see as an ambiguity in Father's teachings.
In my opinion, the comparison in the dialogue between placing a weight on a china plate and dropping the weight onto the plate is slightly misleading. Specifically I would like to consider the statement: "In the first case, the plate has only the pound of weight to sustain; in the other, the weight must be multiplied with the velocity..." In the example of placing the weight on the plate the weight has virtually no momentum due it having almost zero velocity. When the weight is dropped, it does have momentum and seeing as the plate cannot recoil (i.e. convert the weight's momentum into its own) then the energy has to go somewhere else, which causes the plate to deform and shatter. However, the placed weight does exert a force on the plate, which is due to a combination of its mass and the acceleration due to gravity. In a different (more extreme) case, the effects of this force can be seen more clearly. Suppose you take that same pound weight and place it on a single grape; the force of weight itself would catastrophically damage the grape, but not through any action of momentum. This is where we need to introduce the concept of potential energy, which you can think of as stored potential energy. Because energy is also a conserved quantity, the kinetic energy that builds up when you drop a weight has to come from somewhere; this kinetic energy is converted from the potential energy which the object had gained as it was lifted into the air (whoever lifted the object had to expend an equal amount of kinetic energy to raise the object).
Seeing as the distinction between kinetic and potential energy was developed during the 19th century, I think it's unfair to blame The Reverend for misleading his children. He had a fairly good go at explaining why dropped weights break plates.
For reasons which may be due to my paying too much attention to badly researched fiction, I had thought that battering rams had predominantly been used for destroying wooden fortifications whereas stone walls were only attacked directly once cannons became available. This turns out to be complete nonsense, battering rams were most definitely used against stone walls.
The talk of cannons got me curious about muzzle velocities and how realistic it was to have a cannon ball travelling at over 1,000 feet per second. Following a little research into the matter, this seems completely reasonable. I've found references to 18th century cannons firing 24 pound balls with a muzzle velocity well over 1,500 feet per second, this is significantly above the speed of sound. I assume that the first objects that men observed travelling faster than this were the projectiles from these weapons (for comparison, the speed of sound in air at sea level is 1,115 feet per second). The choice of 28 pounds as the mass for an example cannon ball is interesting as it appears to be extremely unusual for cannon balls; I guess that this number was chosen to make the calculated velocity just over a thousand. Cannons firing 24 and 32 pound shot were far more common, although you should feel free to correct me on this if you know something interesting about the history of cannons.
Everything else here meets with my approval. I started to check the mathematics of the momentum of objects dropped from successively increasing heights, I then heroically gave up. I got into a unit conversion nightmare and rather masterfully failed to get the equations right, it also didn't seem to be worth spending that amount of time in order to provide you with what would have been a short and boring paragraph. This may be a good example of why I did not pursue a career in maths.
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