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How would I write a summation for the average speed of an object, given $10$ speeds (which were calculated $r=\frac{d}{t}$)? Would it be:

$$ \sum_{i = 1}^{10} \frac{d_i}{t_i}$$

or am I wrong? I'm not sure about the subscript i's after distance and time, and if they are actually number sequences that I could use. Thanks.

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up vote 2 down vote accepted

No. Generally average speed is defined as the total distance divided by the total time. That would be

$$ \frac{\sum_{i = 1}^{10} d_i}{\sum_{j = 1}^{10} t_j} $$

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This type of thing leads to some amusing puzzles. Like if you take a trip and for the first half you go 30 mph and for the last half you go 70 mph, is your average speed 50 mph? Answer: depends what you mean by half. If "half" means half of the time, then the answer is yes, but if "half" means half of the distance, then the answer is no.

Similarly: if you average 30 mph on an outbound trip, how fast do you have to go on the return trip in order to average 60 mph for the whole trip?

Similarly: construct a scenario where there are two baseball players, and on each day player 1 has a higher batting average than player 2, but after a week player 2 has a higher (cumulative) average than player 1.

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