- #1

CynicusRex

Gold Member

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## Homework Statement

For the first half of a trip a car has velocity v

_{1}; for the second half of a trip it has velocity v

_{2}. What is the mean velocity of the car?

(The book does not mention a direction.)

## Homework Equations

Arithmetic mean: $$\frac{v_{1}+v_{2}}{2}$$

Harmonic mean:

$$\frac{1}{(\frac{1}{v_{1}}+\frac{1}{v_{2}})\frac{1}{2}}$$

## The Attempt at a Solution

[/B]

The velocity is v

_{1}for 1/2 of a trip and v

_{2}for the other 1/2.

How far do they get if they if they were driving at 1 km/h?

$$\frac{v_{1}}{v_{1}}km/h

= 1 km/h \rightarrow

\frac{1}{2v_{1}}trip$$

$$\frac{v_{2}}{v_{2}}km/h

= 1 km/h \rightarrow

\frac{1}{2v_{2}}trip$$

Now, what is the arithmetic mean between those two trip distances they've traveled at 1 km/h? Or, the mean trip distance at 1 km/h is:

$$\frac{\frac{1}{2v_{1}}+\frac{1}{2v_{2}}}{2}=\frac{1}{v_{1}}+\frac{1}{v_{2}} trip$$

But, we need the mean velocity of 1/2 of a trip.

$$\frac{\frac{\frac{1}{v_{1}}+\frac{1}{v_{2}}}{\frac{1}{v_{1}}+\frac{1}{v_{2}}}}{2}=\frac{1}{2}trip\rightarrow v\text{̅}=\frac{1}{(\frac{1}{v_{1}}+\frac{1}{v_{2}})\frac{1}{2}}km/h$$

I know the harmonic mean is the correct solution, but do I get there correctly?