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In math, you say: velocity is the rate of change of position with respect to time.

I'm looking for a better, mathematically correct way to say that, without using the phrase with respect to and velocity is the time-derivative of position.

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Maybe you should ask this on Math.SE? –  Robusto Feb 16 '11 at 18:53
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Since we are talking about mathematically correct ways, it should be noted that velocity is not quite the same as speed. –  RegDwigнt Feb 16 '11 at 19:26
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I wasn't aware anybody suggested otherwise... Velocity is the rate of change of position with respect to time. There's no reason a rate can't also be a vector. dr/dt (r being a position vector) is a derivative and a derivative is, by definition, a rate. –  advs89 Feb 16 '11 at 21:35
    
I agree that it should be asked on math.SE, as it is asking for a mathematically correct way to say a sentence. A mathematically correct way is different from a grammatically correct way. –  kiamlaluno Feb 16 '11 at 23:21
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I disagree. The asker is looking for a better way to say it in English, but, of course, the statement should remain mathematically correct, i.e. retain its current meaning. –  Jimi Oke Feb 17 '11 at 0:42

5 Answers 5

I would just say over:

An object's velocity is the change in its position over time.

Which is a natural expression of the derivative dx/dt.

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+1 That's how I (and my fellow students) tend to say it. –  kitukwfyer Feb 16 '11 at 20:43
    
It's true but not synonymous with the original statement. The "with respect to" means that position is a function of time. Or, in other words, the independent variable in the function is time. I make this distinction because you can't substitute "over" in place of "with respect to" in most cases... only this one. For instance, if I have f(x)=3x^2, you could say "I defined F with respect to x". You would not say, however, "I defined F over x." –  advs89 Feb 16 '11 at 21:29
    
N.B. In mathematics, "dx/dt" is usually read "dx by dt", though, not "over" (which is reserved for actual fractions). –  Neil Coffey Feb 16 '11 at 22:21
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...For the record, I've never heard "dx by dt." I don't think I've seen it in a textbook either. I generally just say/ hear "dx dt." ....And actually I have heard "A function, f, is defined over x." Weird. –  kitukwfyer Feb 16 '11 at 23:09
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@kiamlaluno: It can be. In one-dimensional motion (moving along a straight line) your initial position could be x = 3 ft (from the origin) and your final position could be x = -3 ft (from the origin) and so you moved backwards six feet. Therefore the change in position is negative six feet and if you did that in one second, your velocity would be -6 ft/s. Speed can never be negative. (because negative implies direction) so in that scenario, change in position over time is a vector. (i.e. velocity) –  advs89 Feb 17 '11 at 20:21

Well, you could say "with relation to" or in some cases "as a function of". I'm not quite sure what's wrong with "with respect to" in this case, though.

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+1 for “as a function of”. I was going to propose that. –  Konrad Rudolph Feb 16 '11 at 19:26
    
+1: yes, "as a function of" is the only thing I can think of that can be substituted for "with respect to." I also agree that there is nothing wrong with "with respect to" as long as you know what it means. –  advs89 Feb 16 '11 at 21:31

How about:

Velocity is the rate of change of position with time.

or possibly

An object's velocity is the amount by which its position changes in a given time.

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I would say velocity is the rate at which an object moves, or velocity is the measurement of the rate and direction of change in position of an object.

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I think this definition is a more accurate description of speed rather than velocity. Velocity has a direction component (location change), whereas speed is purely the rate. –  Marcin Feb 16 '11 at 19:40
    
The speed is the scalar component of the velocity; I don't think it is possible to explain the concept of vector in few words. –  kiamlaluno Feb 16 '11 at 20:20
    
@kiamlaluno: I believe he was agreeing with you. He was saying that your definition is better than the others. –  advs89 Feb 16 '11 at 21:39
    
@advs89: I think he is saying that I gave the definition of speed, not velocity. –  kiamlaluno Feb 16 '11 at 23:12
    
Speed, velocity, flow rate with respect to time, growth rate with respect to time... anything other than WRT! –  bobobobo Feb 17 '11 at 0:19

Ok how about as?

Velocity is the rate of change of distance AS TIME changes.

Say we have the volume of a cone

Then say we take the partial derivative with respect to cone radius.

Then we could say the rate of change of that volume AS radius changes, is that equation.

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