## Is velocity a limit?

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jewish_scientist
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### Is velocity a limit?

I thought that velocity and acceleration where /\x / /\t and /\v / /\t. However, I was in my College Physics class and the professor defined velocity as lim /\t->0 /\x / /\t and acceleration as lim /\t->0 /\v / /\t. I would understand why I was not told the exact truth before, but I still do not see why they are limits. Can one of you please explain it to me? Thank you.
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Xanthir
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### Re: Is velocity a limit?

That's the distinction between average velocity/acceleration (your first one) and instantaneous velocity/acceleration (your second one).

(And btw, the character you're looking for is Δ. Try NotOnYourKeyboard for most of the commonest characters.)
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f5r5e5d
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### Re: Is velocity a limit?

there's velocity: the vector, acceleration too - vectors often written in bold: v, a

Neil_Boekend
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### Re: Is velocity a limit?

Δ is a limit to zero. Your college physics teacher was just more verbose.
The way I was taught how to think of actual, momentary acceleration, in this context:
You take a difference in speed. It took some time to achieve that difference in speed. Divide the difference in speed over the time so you get the average acceleration. Now make that difference in speed smaller and smaller until it is almost zero. The time does the same. But you can still calculate the average acceleration by dividing.
Now what Δ does is take that difference in speed and difference in time and interpret them as limits to zero. This gives your actual acceleration, right now. Not the acceleration speed over a distance or a time, but your actual acceleration.
Thus the Δ is a limit to zero. It can not be otherwise.

(The same works with speed by the way)
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### Re: Is velocity a limit?

Δ is not usually a limit as it's taken to zero, it's sometimes even an order 1 change; δ is usually much smaller than order 1 and a d is the limit as it's taken to zero.

The reason we tend to take the limit is that derivatives are much easier to deal with than discrete derivatives (in particular, it's much easier to integrate up a derivative than sum up a discrete derivative).
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Xanthir
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### Re: Is velocity a limit?

Yeah, Δ is just "change in". It's used for any duration of change. As I pointed out above, Δx/Δt is the average velocity. And then if you take the limit toward zero you get the instantaneous velocity.
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### Re: Is velocity a limit?

Velocity is the time-derivative of position, and the derivative of a function is defined as a limit.

As for the notation, eSOANEM pretty much covered it, but it's worth pointing out that the capital letter delta is also used for the Laplacian. That makes it a pretty good idea to explicitly state what you mean when you write down "Δf"!

Xanthir
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### Re: Is velocity a limit?

And again, that's the instantaneous velocity. The OP's first thing was the average velocity, which doesn't need to invoke limits in any way.
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### Re: Is velocity a limit?

True, of course -- but in physics it's rather rare for the word "velocity" (without any specifiers) to refer to anything other than dr/dt (well, sometimes people use it to mean |dr/dt|, but the proper term for that is "speed"). I felt that it was important to point out that the limit has nothing to do with velocity per se, but rather forms the definition of the derivative, which happens to be how (instantaneous) velocity is related to position. I.E. rather than saying "v≡limΔt→0(Δx/Δt), and that happens to coincide with dx/dt", it should be "v≡dx/dt≡limΔt→0([x(t+Δt)-x(t)]/Δt)".

ETA: "x" in this context being the one-dimensional "special case" of the general position vector r, of course.

Neil_Boekend
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### Re: Is velocity a limit?

I should not answer questions on definitions I haven't used in a decade when I am in a hurry. It is a reliable way to generate nonsense that looks scientific and we already have enough of that.
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doogly
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### Re: Is velocity a limit?

Yeah, it's a little awkward in that we start physics 1 by being very careful about the difference between average and instantaneous velocity. It is actually important to understand the conceptual distinction, but then we will never really use average velocity for much at all. Awk.
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### Re: Is velocity a limit?

Average speed is knowing that if you leave now you'll get to grandma's house in 3 hours, taking into account all the traffic lights along the way. Instantaneous speed is looking down at your speedometer and realising that if the police are around you're in big trouble.
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chenille
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### Re: Is velocity a limit?

doogly wrote:It is actually important to understand the conceptual distinction, but then we will never really use average velocity for much at all.

To be fair, when writing equations you pretty much only use instantaneous velocity, but when you are doing physics 1 experiments you often measure an average velocity instead.

doogly
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### Re: Is velocity a limit?

Sure, but then the thread connecting lab to lecture is flimsy at best in physics 1. The kiddies have such difficult times with this.
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jewish_scientist
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### Re: Is velocity a limit?

I remember my high school physics class got in a huge fight because the teacher did not tell us about the difference between instantaneous velocity vs. average velocity. He made a test that asked for the velocity of such and such; but then he only gave credit to people who put the instantaneous velocity. When he said he would give credit to anyone who put the average velocity as the answer, everyone who put the instantaneous velocity wanted to get extra points. People where complaining about it for the couple of weeks.

Xanthir, thank you so much for linking that. I have been driven crazy trying to figure out how to type symbols.
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drachefly
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### Re: Is velocity a limit?

Wow, and this was a physics teacher? The difference between instantaneous and average velocity deserves basically as much time as needed for the teacher to be sure it has soaked in, from a starting point of assuming it hasn't at all.

jewish_scientist
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### Re: Is velocity a limit?

He actually never taught us what instantaneous velocity was, even after that test. Honestly, he was a nice guy, but not the best honor physics teacher. I got the feeling that he had not taught a class before.
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drachefly
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### Re: Is velocity a limit?

Yeah, that's the problem with being bright or otherwise talented - you don't understand what the hard part is.

sciencenerd
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### Re: Is velocity a limit?

Yea, as other people have said, the difference as time approaches 0 is talking about instantaneous velocity, velocity right at that point. Average velocity is simply the difference in time for a given time period.

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