I know to obtain the average of some vectors I compute the mean of the one dimensional components of each vector and constitute the average vector. My question is : why is this untrue for average velocity and acceleration? average velocity is computed as such only when it's one dimensional, whilst average acceleration isn't computed using this method at all، why is this?
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Because time links into it, let me give an example:
We have chosen a coordinate system, and in this coordinate system, we have a mass that only has two velocities (easy), one of them is $v_1=(0,1)$ and the other one is $v_2=(1,0)$. Now what is the average velocity? Essentially it is where the point ended up, divided by the time it got there. But in order to find this out, you need to know how long either velocity was active.
So you need time, so just adding the two vectors together will not work. The average is now $$v = \frac{v_1 \cdot t_1 + v_2 \cdot t_2}{t_1 + t_2}$$
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$\begingroup$ It is unfortunate that textbook definitions of "average velocity" do not indicate that it is a "time-weighted average of velocities", as you show in your example. Most texts just write $\Delta \vec x/\Delta t$, but unfortunately leave out your calculation. In general, it is $\displaystyle \vec v_{avg}=\frac{\int \vec v\ dt}{\int dt}$. $\endgroup$– robphyCommented Jun 30, 2021 at 19:40
v, what do you want the average velocity to be?(0,0,0)or the scalarv? The per-component method gives you(0,0,0). $\endgroup$(0,0,0)will tell you that the object mainly stays in the same place over a long-term observation. The scalar averagev(first take the scalar momentary speeds, then their average) is useful e.g. if this is a car and you want to talk about fuel consumption. $\endgroup$