# No such thing as instantaneous speed?

1. Mar 2, 2015

### Mike1612

I've just come back to physics a decade after school and starting again from the bottom so this might be a very basic, even silly question.

Reading about speed, velocity etc the text talked about instantaneous speed being being different to average speed in that it is the speed of an object at any given instance in time. But wouldn't this idea lead to an equation for speed being s=0/0 since at any given instant distance and time would be 0. Distance cannot be >0 if time is 0 and time cannot be >0 because then any value for s would still be an average.

Have I gone wrong somewhere or is instantaneous speed really just average speed where the change in time is infinitesimally small?

2. Mar 2, 2015

### phyzguy

You should study the process of taking a limit. The instantaneous speed is defined as the limit of the ratio(Δx/Δt) as Δt shrinks to zero. It is possible for this ratio to be well defined even as the two quantities Δx and Δt approach zero.

3. Mar 2, 2015

### Mike1612

I'm watching youtube vids on limits as we speak!

Btw what is x in this instance? Is it distance?

4. Mar 2, 2015

Yes.

5. Mar 2, 2015

Thanks!

6. Mar 2, 2015

### Staff: Mentor

Welcome to calculus!

7. Mar 2, 2015

Staff Emeritus
One way to think about it is the old joke, "Officer, I can't have been going 70 miles an hour. I've only been driving for 20 minutes!"

8. Mar 2, 2015

### samsara15

Can you have instantaneous speed? What about jerk and hyperjerk?

9. Mar 2, 2015

### phyzguy

I knew a guy who was a hyperjerk.

10. Mar 2, 2015

### Staff: Mentor

For how long?

11. Mar 2, 2015

### Staff: Mentor

Post #2 answers the first. Based on that, can you answer the others?

12. Mar 3, 2015

### samsara15

dx/dt is the velocity. A falling object instantly attains an acceleration, dx^2/dt^2, insofar as I know. but if that acceleration were to change, its rate of change, dx^3/dt^3 is the jerk. If post 2 answered that question, it went right over me. I guess the next one up the line is jounce, which I'd never heard of, but apparently, there are applications for these things.

http://en.wikipedia.org/wiki/Jerk_(physics)

Apparently, sometimes even higher derivatives of motion become an issue, but how they would do so escapes me. .

Last edited: Mar 3, 2015
13. Mar 3, 2015

### student34

After just fresh off of studying this topic in university, I must say that this is a very interesting question with a very interesting answer. Here is how I make sense of all of this.

To put it crudely, when you have some d amount of distance over some t amount of time, often either the t or the d will approach 0 a little or a lot faster than the other. But the important thing to know is that they never actually reach zero (except for constant parts of functions (flat areas)).

For the case of accelerating graphs, the variable that loses this race to zero will become infinitesimally small, and the variable that wins the race will also become infinitely small but will have a longer infinitesimal length. Neither quantity makes it to zero. Except for d = t, one quantity will always be infinitesimally smaller than the other . It's mind blowing, at least for me.

Now, if there is no acceleration such as d = 3t, then the work of finding a limit is already done for you in the form of the graph itself. The ratio stays the same as the denominator and numerator go to zero. In other words, if you zoomed in on the infinitesimal of a linear graph, nothing about it should look different. Neither numerator nor denominator will ever get to zero.

Finally, in the case of a constant, the changes in the denominator and numerator are done for you again. As you would notice, the numerator actually makes it to zero because it is zero. And now the the denominator must be some nonzero width.

I only know the basics, first-year stuff. So hopefully if I am wrong about this simplification, someone will correct me.

There seems to be endless amounts of courses one can take on mathematical analysis, and I can't even imagine what is taught at those levels.

Last edited: Mar 3, 2015
14. Mar 3, 2015

### brainpushups

If you feel somewhat bothered and bewildered by the notion of infinitesimals you are not alone. Although mathematicians such as Fermat, Newton, Leibniz, Euler and others made extensive use of infinitesimals there was no rigorous treatment until Cauchy, Weierstrass, and Reimann formulated calculus in terms of limits instead of infinitesimals.

It is interesting to compare the approaches taken by Newton, Leibniz, and Euler. Newton's formulation was completely geometric (his so-called fluents and fluxions) whereas Leibniz' approach was independent of geometry. Leibniz conception of infinite vs. infinitesimal is not unlike how the notion is approached intuitively in physics today. Leibniz might say that the radius of the earth is infinitesimal when it is compared to the distance to the stars.

Euler ran with Leibniz formulation but put in on somewhat more rigorous footing. He expressed the difference between finite and infinitesimal values by writing $Δs ± ds = Δs$ where ds is called a cipher.

In some notes I took a little while ago while I was reading Euler As a Physicist I wrote that the stages of how infinitesimals were conceived was something like this
i) the continuum is assignable points separated by unassignable gaps (Newton)
ii) the continuum is composed of an infinity of indivisible points, smaller than any assignable, with no gaps in between them (Leibniz?)
iii) a line is composed of infinitely many small lines each of which is divisible and proportional to a generating motion at an instant (Euler)
iv) infinitesimals are fictitious which may be used to abbreviate mathematical reasonings and are justifiable in terms of finite quantities taken to be arbitrarily small such that the resulting error is smaller than any pre-assigned margin (Weierstrass I believe)

15. Mar 4, 2015

### yuganes warman

i don't understand how limit works.Explain please thanks

16. Mar 4, 2015

### brainpushups

Read this then let us know if you have questions.

17. Mar 9, 2015

### Mark Harder

If you want to know some more about limits, esp. limits of quotients, look up L'Hopital's law. It's a neat trick. You can find the Limit[x->0] Sin(x)/x easily, for instance. Wikipedia is as good a place as any when you don't have a calculus text handy.

18. Mar 10, 2015

### variation32

Hi Mike1612 (guessing it'that's not year of birth).

You may feel chuffed that you have identified the instantaneous speed problem. Don't think for a moment that it's silly. Contemplating the infinitesimal, deciding whether it is a legitimate mathematical trick or a bold-faced cheat, tormented Newton too. Infinities (both upper and inner) gave pre-Calculus mathematical geniuses vertigo for centuries and held mathematics back until Isaac determined that someone had to make the leap (and get the publication out before Leibniz).

I still consider the ratio of infinitesimals a half-cheat. Imagine we have the quantity x/2x. That would be a half, no matter how big or heavy or whatever "x" is. If x = 1,000 or Ford Transit, "1000/2000" and "Ford Transit/2 Ford Transits" both come to a half. With the one gnarly exception of if x = 0. In which case, is it legitimate to preserve the ratio, the half, even after you have eliminated its constituent components? It's head-spinning stuff, but for Calculus you simply accept that yes, that's how it works and, a few weeks into your studies, you get used to it.

19. Mar 12, 2015

### Mark Harder

The difficult and non-rigorous concept of infinitesimals was abandoned in favor of the more precisely defined notion of the 'limit'. The infinite and the infinitesimal are both considered as the absence of a limiting quantity for some series. In the case of x/2x, the limit as x->0 is the limit of the ratio of the derivatives of x and 2x w.r.t. x: dx/dx over d2x/x, or 1/2. No notion of infinitesimals is required.