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**1. Homework Statement**

(translated from Portuguese) during the performance test of a new automobile model, the pilot goes through the first half of a track with an average velocity of 60 km/h and the second half with 90 km/h. what is the average velocity during the complete test, in km/h?

**2. Homework Equations**

[tex]v = \frac{2v_1v_2}{v_1 + v_2}[/tex]

**3. Attempt at solution**

initially, we weren't sure about how we would solve it; we thought it was through arithmetic average (average v between v1 and v2 = v = (v1+v2)/2).

but it was solved in class, and the teacher said we were supposed to use that special formula, 2*v1*v2/(v1 + v2) in all situations similar to this, that is, when the problem asks us to find the average of velocities when the distance traveled is the same, but with different velocities. look:

since the pilot goes through the same distance (half of the path) with different velocities, v1 = 60 km/h and v2 = 90 km/h, the general formula for this situation is:

[tex]v = \frac{2v_1v_2}{v_1 + v_2} \Rightarrow v = \frac{2(60)(90)}{60 + 90} = \frac{10800}{150} = 72 km/h.[/tex]

our question is: how do we obtain this formula, 2*v1*v2/(v1+v2)? why do we need to use this formula specifically, that is, why can't we just calculate the arithmetic average between 60 and 90, which would be (60 + 90) / 2 = 150 / 2 = 75?

thank you in advance.

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