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In order to get to the subtleties in a clearer fashion, we remind you of a |
Jjoke which you surely must have heard. At the point where the lady in the car |
is caught by a cop, the cop comes up to her and says, “Lady, you were goiỉng |
60 miles an hour!” She says, “hat ”s impossible, sir, I was travelling for only |
seven minutes. It is ridiculous—how can I go 60 miles an hour when Ï wasn't |
goïng an hour?” How would you answer her 1Ý you were the cop? Of course, If |
you were really the cop, then no subtleties are involved; it is very sỉimple: you say, |
“Tell that to the judge!” But let us suppose that we do not have that escape and |
we make a more honest, intellectual attack on the problem, and try to explain to |
this lady what we mean by the idea that she was going 60 miles an hour. .Jjust |
what do we mean? We say, “What we mean, lady, is this: if you kept on going |
the same way as you are going now, in the next hour you would go 60 miles.” She |
could say, “Well, my foot was off the accelerator and the car was sÌlowing down, |
so 1f Ï kept on going that way it would not go 60 miles” Ôr consider the falling |
ball and suppose we want to know its speed at the time three seconds ïf the ball |
kept on going the way it is going. What does that mean——kept on øccelerating, |
--- Trang 165 --- |
going faster? No—kept on goiỉng with the same øeloc#u. But that is what we are |
trying to definel For if the ball keeps on going the way it is goïng, it will just |
keep on going the way i% is going. Thus we need to defñne the velocity better. |
'What has to be kept the same? 'Phe lady can also argue this way: “If I kept on |
going the way m goiïng for one more hour, Ï would run into that wall at the end |
of the streetl” It is not so easy to say what we mean. |
Many physicists think that measurement is the only defnition of anything. |
Obviously, then, we should use the instrument that measures the speed——the |
speedometer——=and say, “Look, lady, your speedometer reads 60.” So she says, |
“My speedometer is broken and didn”t read at all” Does that mean the car is |
standing still? We believe that there is something to measure before we build |
the speedometer. Only then can we say, for example, “The speedometer isn'$ |
working right,” or “the speedometer is broken.” That would be a meaningless |
sentence ïf the velocity had no meaning independent of the speedometer. So we |
have in our minds, obviously, an idea that is independent of the speedometer, |
and the speedometer is meant only to measure this idea. So let us see IŸ we can |
get a better defnition of the idea. We say, “Yes, of course, before you went an |
hour, you would hít that wall, but if you went one second, you would go 88 feet; |
lady, you were going 88 feet per second, and ïf you kept on going, the next second |
it would be 88 feet, and the wall down there is farther away than that.” She says, |
“Ves, but there's no law against going 88 feet per secondl 'There ¡is only a law |
against going 60 miles an hour.” “But,” we reply, “it's the same thing.” If it 2s |
the same thing, it should not be necessary to go into this cireumlocution about |
88 feet per second. In fact, the falling ball could not keep going the same way |
even one second because it would be changing speed, and we shall have to delne |
speed somehow. |
Now we seem to be getting on the right track; it goes something like this: If |
the lady kept on goïing for another 1/1000 of an hour, she would go 1/1000 of |
60 miles. In other words, she does not have to keep on going for the whole hour; |
the point is that for ø mmornent she is goïng at that speed. Now what that means |
is that if she went just a little bit more in time, the extra distance she goes would |
be the same as that of a car that goes at a s‡eady speed of 60 miles an hour. |
Perhaps the idea of the 88 feet per second is right; we see how far she went In |
the last second, divide by 88 feet, and ïf it comes out 1 the speed was 60 miles |
an hour. In other words, we can fnd the speed in this way: We ask, how far |
do we go in a very short time? We divide that distance by the time, and that |
gives the speed. But the time should be made as short as possible, the shorter |
--- Trang 166 --- |
the better, because some change could take place during that time. If we take |
the time of a falling body as an hour, the idea is ridiculous. lf we take it as a |
second, the result is pretty good for a car, because there is not much change in |
speed, but not for a falling body; so in order to get the speed more and more |
accurately, we should take a smaller and smaller time interval. What we should |
do is take a millionth oŸ a second, and divide that distance by a millionth of a |
second. “The result gives the distance per second, which is what we mean by the |
velocity, so we can defñne it that way. That is a successful answer for the lady, or |
rather, that is the defnition that we are going to se. |
The foregoing defñnition involves a new idea, an idea that was not available to |
the Greeks in a general form. That idea was to take an ?nfinitesimal distance and |
the corresponding ?nfin2tesimal time, form the ratio, and watch what happens |
to that ratio as the time that we use gets smaller and smaller and smaller. In |
other words, take a limit of the distance travelled divided by the time required, |
as the time taken gets smaller and smaller, ød ?nfimuitưm. 'Phis idea was invented |
by Newton and by Leibniz, independently, and is the beginning of a new branch |
of mathematics, called the djferential calculus. Calculus was invented in order |
to describe motion, and its first application was to the problem of defning what |
is meant by going “60 miles an hour.” |
Let us try to defne velocity a little better. Suppose that in a short time, c, |
the car or other body goes a short distance z; then the velocity, 0, is defned as |
U = đc, |
an approximation that becomes better and better as the is taken smaller and |
smaller. If a mathematical expression ¡is desired, we can say that the velocity |
cquals the limit as the is made to go smaller and smaller in the expression #/c, |
ø = lim `, (8.3) |
ec>0 € |
We cannot do the same thing with the lady in the car, because the table is |
Iincomplete. We know only where she was at intervals of one minute; we can |
get a rough idea that she was going 5000 ft/min during the 7th minute, but we |
do not know, at exactly the moment 7 minutes, whether she had been speeding |
up and the speed was 4900 ft/min at the beginning of the 6th minute, and is |
now 5100 ft/min, or something else, because we do not have the exact details in |
between. So only If the table were completed with an infnite number oŸ entries |
could we really calculate the velocity from such a table. On the other hand, |
--- Trang 167 --- |
when we have a complete mathematical formula, as in the case of a falling body |
(Eaq. 8.1), then it is possible to calculate the velocity, because we can calculate |
the position at any time whatsoever. |
Let us take as an example the problem of determining the velocity of the |
falling ball at the particular time 5 seconds. Ône way to do this is to see from |
Table 8-2 what it dịd in the 5th second; it went 400 — 256 = 144 Ít, so ï is goïng |
144 ft/sec; however, that is wrong, because the speed is changing; øn the œuerage |
1€ is 144 ft/sec during this interval, but the baill is speeding up and is really goïng |
faster than 144 ft/sec. We want to lnd out ezactflU hou ƒast. The technique |
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