Question

The Batmobile travels 30 miles at an average speed of 60 mi/hr and another 30 miles at an average speed of 30 mi/hr. What is the average speed of the car over the 60 miles? The answer is 40 m/s. How do I show my work?

Answer 1

We are basically finding the average of several averages. You can do this by just adding the averages and dividing by the number of averages, but this will only work if all of the averages are equally weighted.

However, I'll show the process of breaking down the weighting of the averages so that it will work no matter what the weightings of the averages are.

Average speed is defined as:\[\overline s=\frac{d}{t}\]Where d is the distance covered over some time interval, t. Let's rewrite these for t so that we can find the time interval for each section of the trip:\[t=\frac{d}{s}\]The time it took for the first part of the trip was:\[t=\frac{30mi}{60\frac{mi}{hr}}=0.5hr\]And the second part:\[t=\frac{30mi}{30\frac{mi}{hr}}=1hr\]Now we can go back to the original average speed equation:\[\overline s=\frac{d}{t}\]We know that the total distance covered is 60 miles (30mi + 30mi) and that the total time it took was 1.5hr (0.5hr + 1hr), so we can plug those numbers into this equation to find the overall average speed:\[\overline s = \frac{d}{t}=\frac{60mi}{1.5hr}=40\frac{mi}{hr}\]Make sense?

Answer 2

thanks a bunch! but how do i get it to be 40 m/s?

Answer 3

It's not 40 m/s, but we can convert 40 mi/hr into m/s using dimensional analysis:\[40\frac{mi}{hr} \times (\frac{1hr}{3600s}) \times (\frac{1609m}{1mi})=17.9\frac{m}{s}\]As you can see, it is not 40 m/s, but 17.9 m/s.

Answer 4

Oh thank you!

Answer 5

That's a very nice explanation. There is also another formula you can use for this particular type of problem:\[2\frac{V_1V_2}{V_1+V_2}\]\[2\frac{(30)(60)}{30+60}=40\]

Answer 6

The equation that Shane_B presents is only valid under very specific circumstances: that all the speed measurements are taken over the same distance. This is the "uneven weighting" issue I described earlier.

You can only use that equation if you know that each speed measurement was taken over the same distance, n.

Answer 7

Thanks...I should have probably clarified that. :)