# Why total time taken s₁/v₁ + s₂/v₂ + s₃/v₃ in average speed?

#### Indranil

As we know the Average speed = Total distance / Total time but why in the case below
Distance traveled = s₁ + s₂ + s₃ and total time taken = s₁/v₁ + s₂/v₂ + s₃/v₃ How is it possible? Could you explain please? so, altogether
Vav = s₁ + s₂ + s₃ / ( s₁/v₁ + s₂/v₂ + s₃/v₃)

#### benit13

speed = distance / time

Therefore,
time = distance/speed

Let's assign some symbols to these quantities...

time = t
speed = v (from 'velocity')
distance = s (don't ask... people just like using s for distance. it crops up a lot in maths too for path lengths and things like that)

The equation above described in symbols is:
$$\displaystyle t = \frac{s}{v}$$

Let's use the numbers 1, 2 and 3 to designate the three parts of a journey. These are called "indices".

Time for first part of journey:
$$\displaystyle t_1 = \frac{s_1}{v_1}$$

Time for second part of journey:
$$\displaystyle t_2 = \frac{s_2}{v_2}$$

Time for third part of journey:
$$\displaystyle t_3 = \frac{s_3}{v_3}$$

Total time for journey:
$$\displaystyle t_{total} = t_1 + t_2 + t_3$$

Substitute for individual times, we get:
$$\displaystyle t_{total} = \frac{s_1}{v_1} + \frac{s_2}{v_2} + \frac{s_3}{v_3}$$

The total distance is:
$$\displaystyle s_{total} = s_1 + s_2 + s_3$$

Therefore, the average speed is:
$$\displaystyle \bar{v} = \frac{s_{total}}{t_{total}} = \frac{s_1 + s_2 + s_3}{\frac{s_1}{v_1} + \frac{s_2}{v_2} + \frac{s_3}{v_3}}$$

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