I Problem in understanding instantaneous velocity

[Mike_bb]

Hello!

I have difficulty in understanding an instantaneous velocity. Some books say that average velocity more and more approximate instantaneous velocity if time interval approaches to zero. Why is it so?

If point has no length then how does average velocity approximate instantaneous velocity at point?

Thanks.

 

[fresh_42]

I read your question as the process that uses secants to approach a tangent.

The point where the tangent is drawn is the instantaneous velocity, and the slope of the secant is the average velocity. If you let $x_1$ approach $x_0,$ which are moments in time, you finally receive the (yellow) tangent at $x_0.$ In this sense, the average (red secants) approaches the instantaneous (yellow tangent) velocity.

The mathematical background is the definition of differentiability. If the velocity curve $f(x)$ (speed at a certain point in time) is differentiable, then you can write
$$
f(x_0+a)=f(x_0)+f'(x_0)\cdot a + r(a)
$$
where $a$ is a very short timespan that can be arbitrarily short. The other terms mean:
\begin{align*}
f(x_0+a)\quad&\text{instantaneous velocity at time } x_0+a\\
f(x_0)\quad&\text{instantaneous velocity at time } x_0\\
f'(x_0)\cdot a\quad&\text{average velocity measured measured during } a\\
r(a)\quad&\text{very small correction term}\\
\end{align*}
$f'(x_0)$ is the slope of the tangent at $x_0$ (instantaneous velocity) derived from the slope of the secants $\dfrac{f(x_0+a)-f(x_0)}{a}$ (average velocity during $a$). The descriptions you mentioned try to describe this formula without using it.

 

[PeroK]

Hello!

I have difficulty in understanding an instantaneous velocity. Some books say that average velocity more and more approximate instantaneous velocity if time interval approaches to zero. Why is it so?

If point has no length then how does average velocity approximate instantaneous velocity at point?

Thanks.

This seems to me perfectly intuitive. If instantaneous anything has any meaning it must be the limit of smaller and smaller averages. What else is it going to be?

You need calculus to formalise this idea, but we should know intuitively what the mathematics is trying to achieve.

 

[TensorCalculus]

Hello!

I have difficulty in understanding an instantaneous velocity. Some books say that average velocity more and more approximate instantaneous velocity if time interval approaches to zero. Why is it so?

If point has no length then how does average velocity approximate instantaneous velocity at point?

Thanks.

One, very unmathematical (and a bit imprecise), way to think about instantaneous velocity intuitively is by thinking of the speed the speedometer in a car shows - that's the instantaneous speed at that point in time. If you can make intuitive sense of the speed readings on a car - then hopefully it shouldn't seem too weird that you can in fact find speed at a particular point in time.

Mathematically, instantaneous velocity is defined as the limit of the average velocity over a time interval as that interval becomes arbitrarily small. We never divide by a time interval of exactly zero, or look at the average velocity over a point with no length — instead, we study what happens as the interval shrinks closer and closer to zero, never quite touching it, and that's why we can define motion at an instant!

 

[Gavran]

The average velocity is the rate of change of position over a time interval, while the instantaneous velocity is the rate of change of position at a point in time. As a time interval approaches a point in time, the average velocity approaches the instantaneous velocity.

 

[Herman Trivilino]

I have difficulty in understanding an instantaneous velocity. Some books say that average velocity more and more approximate instantaneous velocity if time interval approaches to zero. Why is it so?

If point has no length then how does average velocity approximate instantaneous velocity at point?

Can you understand the slope of a line that is tangent to a curve at a point? This is the same issue. How can a point, which has no length, have a slope?

 

[weirdoguy]

Mathematically, instantaneous velocity is defined as the limit

Yes, and I would say you would do yourself a huge favour if you'd learn what limit is, mathematically. I know that the definition is overwhelming (it took me a lot of time to grasp it), but once you understand it, a whole new world of wonders opens up

 

[Herman Trivilino]

Can you understand the slope of a line that is tangent to a curve at a point? This is the same issue. How can a point, which has no length, have a slope?

So do you understand that you choose a nearby point and use those two points to determine a slope. Then you choose another point that's closer yet, and repeat. Keep repeating. You will find that the slope converges to a value, and that value is the slope at that point.

All of this is made rigorous using calculus.

 

[Mike_bb]

So do you understand that you choose a nearby point and use those two points to determine a slope. Then you choose another point that's closer yet, and repeat. Keep repeating. You will find that the slope converges to a value, and that value is the slope at that point.

All of this is made rigorous using calculus.

Limit allows us to make jump from the secant to the tangent.

 

[Gavran]

Limit allows us to make jump from the secant to the tangent.

No, it does not.
The limit means that the secant line approaches the tangent line, but it never becomes the tangent line.

 

[Mike_bb]

No, it does not.
The limit means that the secant line approaches the tangent line, but it never becomes the tangent line.

No. Limit doesn't mean the process of approaching to the tangent line itself. Limit is just a number (coefficient of tangent line slope).

 

[weirdoguy]

but it never becomes the tangent line.

No, the limit of those lines is the tangent line. You are confusing limit with the sequence and whether or not it's part of the sequence. For numerical sequences limit is a number, that may or may not be a part of the sequence. But that doesn't matter. It is a number, not a process. Limits do not approach anything, they just are. The sequence 1/n approaches 0 as n grows, yes, but the limit is 0, not approach 0.

Btw, I love how selective memory loss is. I do remember that in topological spaces that are Hausdorff, there is only one limit, but I do not remember what Hausdorff spaces are

 

[Svein]

Copyied from "Real Analysis" (H.L. Royden):
These are topological spaces.
T_{1}: Given two distinct points x and y, there is an open set which contains y but not x.
T_{2}: Given two distinct points x and y, there are disjoint open sets O_{1} and O_{2} such that x\in O_{1} and y\in O_{2}.
T_{3}: In addition to T_{1}:, given a closed set F and a point x not in F there are disjoint open sets O_{1} and O_{2} such that x\in O_{1} and F\subset O_{2}.
T_{4}: In addition to T_{1}:, given two disjoint closed sets F_{1} and F_{2}, there are disjoint open sets O_{1} and O_{2} such that F_{1}\subset O_{1} and F_{2}\subset O_{2}

Spaces that satisfy T_{2} are called Hausdorff spaces
Spaces that satisfy T_{3} are called regular spaces
Spaces that satisfy T_{4} are called normal spaces

 

[Gavran]

By “it never becomes the tangent line” in the post #10 I mean that the secant line never becomes the tangent line.

 

[PeroK]

I suspect we might be arguing about words here.

 

[Mike_bb]

By “it never becomes the tangent line” in the post #10 I mean that the secant line never becomes the tangent line.

See answer
"No, the limit of those lines is the tangent line. "

Do you agree with this answer?

 

[fresh_42]

By “it never becomes the tangent line” in the post #10 I mean that the secant line never becomes the tangent line.

That depends on the case. $r(a)\equiv 0$ is not ruled out. Otherwise, you are simply insisting on replacing "becomes" with "approaches". That's nitpicking. Given that the question itself is settled in an imprecise wording, this is not really helpful.

 

[Mike_bb]

One, very unmathematical (and a bit imprecise), way to think about instantaneous velocity intuitively is by thinking of the speed the speedometer in a car shows - that's the instantaneous speed at that point in time. If you can make intuitive sense of the speed readings on a car - then hopefully it shouldn't seem too weird that you can in fact find speed at a particular point in time.

I read about speedometer but I have a doubt. Speedometer has finite number of generated impulses (per second or per 1meter) but on the time interval [T1;T2] we have infinite number of instantaneous speeds.

Could you explain how does speedometer work? Does speedometer show exact instantaneous speed at each time moment? How is it possible? Thx.

 

[PeroK]

I read about speedometer but I have a doubt. Speedometer has finite number of generated impulses (per second or per 1meter) but on the time interval [T1;T2] we have infinite number of instantaneous speeds.

Could you explain how does speedometer work? Does speedometer show exact instantaneous speed at each time moment? How is it possible? Thx.

It's not possible. Instantaneous velocity is part of the mathematical model. In Newtonian physics you can postulate that a particle has a physical instantaneous velocity, if you want to. Or, you can be more agnostic and say you don't care about whether instantaneous velocity corresponds to "an element of physical reality", if I can put it like that.

In any case, eventually Quantum Mechanics enters the picture, and your classical elements of physical reality are seen to be not so physically fundamental after all. That's why you are best treating all such concepts as part of the mathematical model, IMO.

This, however, is really the technicalities of how we do physics.

All that said, it's difficult for me to comprehend why instantaneous velocity is a problem. If someone says a F1 car was doing 185mph when it crossed the finishing line, it's hard for me to imagine the life experiences you have had that makes that statement unintuitive?

 

[fresh_42]

I read about speedometer but I have a doubt. Speedometer has finite number of generated impulses (per second or per 1meter) but on the time interval [T1;T2] we have infinite number of instantaneous speeds.

Could you explain how does speedometer work? Does speedometer show exact instantaneous speed at each time moment? How is it possible? Thx.

They measure how often the wheels spin.
https://en.wikipedia.org/wiki/Speedometer#Operation

 

[Gavran]

See answer
"No, the limit of those lines is the tangent line. "

Do you agree with this answer?

I agree that the limit of those lines is the tangent line, but I do not agree with your statement in the post #9. That is like stating that $$ \lim_{x\to\infty}x=\infty $$ allows us to make a jump from a real number to infinity.

 

[PeroK]

I agree that the limit of those lines is the tangent line, but I do not agree with your statement in the post #9. That is like stating that $$ \lim_{x\to\infty}x=\infty $$ allows us to make a jump from a real number to infinity.

This is one example where the concept of infinity can be introduced in mathematics. In this case, $\pm \infty$ become an extension of the Real numbers, representing part of the set of well-defined limits of a Real sequence.

The set of well-defined limits becomes the extended Real line: $\mathbb R \cup \{\pm \infty \}$

https://en.wikipedia.org/wiki/Extended_real_number_line

 

[willyengland]

I think this is one of the many problems that arise when the purely theoretical concept of infinity meets reality.In reality, there is no real infinity. Everything is finite.
For example, a car, of course, has an instantaneous speed, but you can't measure it. Only approximately.
The whole thing is quite curious and goes deep into the philosophical realm.
Perhaps everything "uniform" is actually quantized.

 

[PeroK]

I think this is one of the many problems that arise when the purely theoretical concept of infinity meets reality.In reality, there is no real infinity. Everything is finite.

There is no problem if you understand the role of mathematics in physics. Derivatives and differential equations are fundamental. You can't throw your arms in the air everytime you see a derivative.

 

[fresh_42]

I think this is one of the many problems that arise when the purely theoretical concept of infinity meets reality.In reality, there is no real infinity. Everything is finite.

There is no perfect circle in the real world, either, but the concept is helpful when calculating gears, orbits, sines, and cosines. I remember that my professor of the differential equation systems class said that there is no real continuity in the world since there is a lot of space between the molecules, or between the atomic nuclei and the electron shells, yet, he taught differential equations. There will always be a gap between reality and mathematics, but are you really worried about what is going on at 0.0000000001 meters when doing classical physics?

 

[willyengland]

There will always be a gap between reality and mathematics

Yes, I just wanted to point that out.

I agree that mathematics is fundamental for our understanding and our models. But there will always be this gap. Possibly this is also fundamental, there are no irrational numbers in nature?

 

[fresh_42]

Yes, I just wanted to point that out.

I agree that mathematics is fundamental for our understanding and our models. But there will always be this gap. Possibly this is also fundamental, there are no irrational numbers in nature?

To put it with Kronecker: "God made the integers, man made the rest."

My personal opinion is that only the natural numbers $1,2,3,\ldots$ are somehow literally natural and even $0$ is a cultural achievement. Someone in India, some 5,000 years ago, had the brilliant idea to count something that wasn't there! More prosaic was it likely an accountant who wanted to improve his balance sheets.

This shows two things. Mathematics is an ideal concept and the question of being real, or natural, or whatever, is a pure philosophical one. And mathematics has always been used to calculate things in the real world. If it helps, we will use it.

 

[PeroK]

My personal opinion is that only the natural numbers $1,2,3,\ldots$ are somehow literally natural and even $0$ is a cultural achievement. Someone in India, some 5,000 years ago, had the brilliant idea to count something that wasn't there! More prosaic was it likely an accountant who wanted to improve his balance sheets.

If one farmer has 20 sheep and 5 goats, and another farmer has 50 sheep and no goats, then (it seems to me) this is already a simple, practical example of why 0 is a natural number. The second farmer can't simply deny the existence of goats, per se.

 

[Mark44]

For example, a car, of course, has an instantaneous speed, but you can't measure it.

Sure you can, and speedometers do exactly this. One kind of speedometer on a motorcycle I own has a gear driven sensor on the front wheel. The sensor has a worm gear that turns when the wheel turns. The worm gear drives a cable, the other end of which causes a magnet to rotate that in turn causes a needle to sweep to a certain position that corresponds to the speed of the motorcycle (speed = magnitude of velocity). Another motorcycle I have has a cable that comes from the transmission that drives the speedometer in a similar way. These measured values are the instantaneous speeds.

 

[gmax137]

There are other speedometers that have a built in "clock." These "chronometric" speedos count how many turns of the wheel occur as the clock counts down, and use this to position the needle. So if you're accelerating, the needle jumps up in speed as the clock times out. I'm going 40, no 50, no 60...

 

[jbriggs444]

Sure you can, and speedometers do exactly this. One kind of speedometer on a motorcycle I own has a gear driven sensor on the front wheel. The sensor has a worm gear that turns when the wheel turns. The worm gear drives a cable, the other end of which causes a magnet to rotate that in turn causes a needle to sweep to a certain position that corresponds to the speed of the motorcycle (speed = magnitude of velocity). Another motorcycle I have has a cable that comes from the transmission that drives the speedometer in a similar way. These measured values are the instantaneous speeds.

The needle has mass and is damped. Undamped needles bounce. A torque that acts on such a needle will not result in an instantaneous change in the needle's position. What the speedometer shows is some sort of approximate weighted average of past velocities.

 

[Mike_bb]

The needle has mass and is damped. Undamped needles bounce. A torque that acts on such a needle will not result in an instantaneous change in the needle's position. What the speedometer shows is some sort of approximate weighted average of past velocities.

I read more about speedometers and I agree with you. Only analogue speedometer with the arrow can show exact value of instantaneous speed because such speedometer is based on induction principle.

 

[Mark44]

The needle has mass and is damped.

Only analogue speedometer with the arrow can show exact value of instantaneous speed because such speedometer is based on induction principle.

The speedometers on my old bikes are strictly analogue, and as far as I know, aren't damped.

 

[robphy]

Here's how I visually motivate the interpretation of the velocity (as the slope of the tangent line) to my students.
Assuming a nice x-vs-t graph,
at the instant T of interest, centered at (T,x(T)),
I zoom in enough so that "my graph looks like a straight-line" in the viewport.
Since the motion is practically a steady-velocity motion for a sufficiently-short time-interval,
the [instantaneous] velocity is practically equal to the slope of that [approximate] straight-line in the viewport.

In the Desmos visualization below,
I have already activated the tangent-line (which you can disable by clicking on the filled circle for its folder).
To animate the zoom, click on the play-button ⏵ of the z-slider. (You can drag the z-slider to manual control the zoom.)

www.desmos.com/calculator/ghfds0lbht