You drive on interstate 10 from san antonio to houston, half the time at 55 km/h and the other half at 90 km/h. on the way back you travel half the distance at 55 km/h and the other half at 90 km/h. what is your average speed
(a) from san antonio to houston,
(b) from houston back to san antonio, and
(c) for the entire trip?
(d) what is your average velocity for the entire trip?
(e) sketch x versus t for (a), assuming the motion is all in the positive x direction.
indicate how the average velocity can be found on the sketch.

 Not the mean of the two speeds. The average speed is the totality of the distance divided by the totality of the time. For convenience, let us say that the distance is 1000 miles. Then the total time is 

(500 / 55) + (500 / 90) = 14.65 hours 

The average speed is 1000 divided by the total time, 68.28 mph. 

The reason that the true average speed, 68.28, comes out lower than Mirt's naive value is that you spend more TIME driving at 55 miles per hour than at 90 miles per hour. That reduces that true average speed below the mean of the two speeds. 

Here's one for Mirt: let's say you want to complete a trip at an average speed of 60 miles per hour. You go the first half of the distance at 30 miles per hour. How fast would you have to go the second half? (Mirt would say 90, but that is not correct. There is in fact no speed for the second half that could raise your average to 60. Let's say the whole trip is 60 miles and you would like to complete it in one hour, for an average of 60 miles per hour. If you do the first half, 30 miles, at 30 miles per hour, you have used up your whole time and cannot raise your average to 60 no matter how fast you go.)