Speed/velocity as a derivative

[rudransh verma]

[Note: Link to the quote below has been pasted in by the Mentors -- please always provide attribution when quoting another source]

https://www.feynmanlectures.caltech.edu/I_08.html

The foregoing definition involves a new idea, an idea that was not available to the Greeks in a general form. That idea was to take an infinitesimal distance and the corresponding infinitesimal 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 traveled divided by the time required, as the time taken gets smaller and smaller, ad infinitum. This idea was invented by Newton and by Leibniz, independently, and is the beginning of a new branch of mathematics, called the differential calculus. Calculus was invented in order to describe motion, and its first application was to the problem of defining what is meant by going “60 miles an hour.”

Let s=16t^2 and we want to find speed at 5 sec.
s = 16(5.001)2 = 16(25.010001) = 400.160016 ft.

In the last 0.001 sec the ball fell 0.160016 ft, and if we divide this number by 0.001 sec we obtain the speed as 160.016 ft/sec. That is closer, very close, but it is still not exact. It should now be evident what we must do to find the speed exactly. To perform the mathematics we state the problem a little more abstractly: to find the velocity at a special time, $t_0$, which in the original problem was 5 sec.

Now the distance at $t_0$, which we call $s_0$, is $16{t_0}^2$, or 400 ft in this case. In order to find the velocity, we ask, “At the time ${t_0}$+ (a little bit), or {t_0} + ε, where is the body?” The new position is $16({t_0} + ε)^2 = 16{t_0}^2+ 32{t_0}ε + 16ε2$.

So it is farther along than it was before, because before it was only 16t20. This distance we shall call s0 + (a little bit more), or${s_0}$ + x (if x is the extra bit). Now if we subtract the distance at t0 from the distance at t0 + ε, we get x, the extra distance gone, as x = 32t0 · ε + 16ε2. Our first approximation to the velocity is
$v = x/ε = 32t0 + 16ε$

The true velocity is the value of this ratio, x/ε, when ε becomes vanishingly small. In other words, after forming the ratio, we take the limit as ε gets smaller and smaller, that is, approaches 0. The equation reduces to,
v (at time ${t_0}$) = $32{t_0}$

In our problem, t0 = 5 sec, so the solution is v = 32×5 = 160 ft/sec. A few lines above, where we took ε as 0.1 and 0.001 sec successively, the value we got for v was a little more than this, but now we see that the actual velocity is precisely 160 ft/sec.

My question is as epsilon reduces to vanishingly small is it really the velocity at t0 or in very small time interval epsilon ? How can you confidently say that it’s the Velocity at ${t_0}$ ?

 

[Arjan82]

The point is that epsilon is infinitesimally small. This means (sort of intuitively) smaller than any positive number that you can think of, but not zero. This effectively means that the statement "the velocity is x at t0" is really equivalent to saying "the velocity is x on average at an infinitesimally small interval around t0".

That is the best answer I can come up with, but admittedly it is a bit hard to wrap your head around it if you encounter it for the first time. It is something that I have found you need to get used to.

 

[jbriggs444]

My question is as epsilon reduces to vanishingly small is it really the velocity at t0 or in very small time interval epsilon ? How can you confidently say that it’s the Velocity at ${t_0}$ ?

That is not a question about physics. That is a question about mathematics.

In physics, we cannot measure arbitrarily small time intervals. Nor can we measure arbitrarily small distances. So the question of what velocity really is does not arise. We know what it approximately is. That is good enough.

In mathematics we define things so that the velocity is not the ratio of any particular small distance to any particular small time. Instead, it is the limit approached by that ratio as the distances and times get smaller and smaller. We confidently say this because we have defined it to be so.

Fortunately, the mathematics generally matches the physics to within experimental accuracy.

 

[rudransh verma]

In physics, we cannot measure arbitrarily small time intervals. Nor can we measure arbitrarily small distances. So the question of what velocity really is does not arise.

That is why we take the help of mathematics.

 

[rudransh verma]

This means (sort of intuitively) smaller than any positive number that you can think of, but not zero. This effectively means that the statement "the velocity is x at t0" is really equivalent to saying "the velocity is x on average at an infinitesimally small interval around t0".

Feels right! something of a average thing. But we do take the value of ε as zero and eliminate ε. I think both of the things are right. If you get infinitely small I believe it doesn’t matter if ε is there or not. And If you see no one accepts one definition. I think it’s something we have to accept since the great Newton discovered it and maths does agree with the experiments.

I think we have to believe that if v is the velocity at some infinitely small ε time interval then it will also be at some point very next to it, ${t_0}$.

 

[jbriggs444]

Feels right! something of a average thing. But we do take the value of ε as zero and eliminate ε. I think both of the things are right. If you get infinitely small I believe it doesn’t matter if ε is there or not. And If you see no one accepts one definition. I think it’s something we have to accept since the great Newton discovered it and maths does agree with the experiments.

I think we have to believe that if v is the velocity at some infinitely small ε time interval then it will also be at some point very next to it, ${t_0}$.

"Feels right" is good for intuition and I have no problem with this intuition. However that is not how mathematics is done. In the standard analysis, a derivative is a limit, not a ratio. There is no debate about this. It is the standard and well accepted definition.

We should probably leave Robinson's non-standard analysis and the transfer principle for when your mathematical education has progressed further. As I understand the transfer principle, a standard limit of ratios (if it exists) is guaranteed to match a non-standard ratio of infinitesimals. Intuitively you might think of this as saying that your intuition was well founded after all.

 

[Mister T]

My question is as epsilon reduces to vanishingly small is it really the velocity at t0 or in very small time interval epsilon ?

If epsilon is zero then there is no very small time interval epsilon.