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I'm pretty sure this has more to do with fundamental Math than Physics and that is why I'm asking this here rather than Physics.SE

Imagine some object travelling along a straight path from point $A$ to $C$
with an initial speed of $u$ m/s. Halfway through it's motion, at point $B$,
it changes it's speed to $v$ m/s. Sure, I know that $$\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{x_1 + x_2 + ... + x_n}{t_1 + t_2 + ... + t_n} = \frac{\sum\limits_{i=1}^n x_i}{\sum\limits_{i=1}^n t_i}$$ I want to find it's average speed in terms of it's initial and final speeds but as I think about it, I realize that there's more than one way to skin this cat.

Time Perspective:
Let $t$ be the total time taken. $$ x_1 = u\cdot (\frac{t}{2})$$ $$ x_2 = v\cdot (\frac{t}{2})$$ $$ \sum x = \frac{t}{2} (u + v)$$ $$ \text{Avg Speed} = \frac{\frac{t}{2} (u + v)}{t} = \frac{u + v}{2}$$

Distance Perspective:
Let $x$ be the total distance. $$ t_1 = \frac{\frac{x}{2}}{u}$$ $$ t_2 = \frac{\frac{x}{2}}{v}$$ $$ \sum t = \frac{\frac{x}{2}}{u} + \frac{\frac{x}{2}}{v} = \frac{\frac{x}{2}(u+v)}{uv}$$ $$\text{Avg Speed} = \frac{x}{\frac{x(u+v)}{2uv}} = \frac{2uv}{u + v}$$

So, as you can see, by looking at the same thing from diferent perspectives, I arrive at two diferent conclusions. Please help me understand why this is so. Which one is the right one and why am I arriving at two different conclusions? If I want to determine the average speed of an object with variable speeds at different points, which formula do I use?

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    $\begingroup$ You need to know whether "halfway through it's motion" [sic] means that the time from $A$ to $B$ is the same as the time from $B$ to $C$, or that the distance $AB$ is the same as the distance $BC$. $\endgroup$
    – mjqxxxx
    Nov 10, 2013 at 18:15

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These are two different scenarios, not two different perspectives. In the first scenario, you are spending half of you time at one velocity and half of your time at the other velocity. In the second scenario you are traveling half the distance at one velocity and traveling half the distance at the other velocity.

For example, if the distance is 300 m, u = 10 m/s and v = 20 m/s

Scenario 1 you will spend 10 sec at 10 m/s and 10 sec at 20 m/s you will travel 100 m in the first 10 seconds and 200 meters in the second 10 seconds. So you traveled 300 m in 20 seconds.

Scenario 2 you travel 150 meters at 10 m/s and 150 meters at 20 m/s It will take 15 seconds to travel the first distance and 7.5 s to travel the second distance. So you traveled 300 meters in 22.5 seconds.

"Halfway through it's motion, at point B," is a little vague but I think they mean scenario two.

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