iSoul In the beginning is reality

# Velocity puzzle

A number of word problems involve vehicle or aircraft speeds over two distances or two time periods and ask what the average speed is. The student is expected to understand the difference between the space-mean speed and the time-mean speed (though these terms are not typically used).

What about the “average velocity”? Since velocity is a vector, is the average velocity the velocity of the resultant motion (displacement), with its magnitude (speed) and direction? Or does it mean the total distance traveled divided by the total travel time — but for what direction?

Say a vehicle travels east for 10 miles in 20 minutes, then travels north for 17 miles in 15 minutes. What is the average velocity? Here are two answers:

(1a) The total distance traveled is 10 + 17 miles, or 27 miles. The total travel time is 20 + 15 minutes, or 35 minutes. The average speed would be 27/35 miles per minute, or about 46 mph.

(1b) Then what is the direction? Take the direction of the displacement velocity. The vehicle travels east 10 miles in 20 minutes, or 30 mph. Then it travels north 17 miles in 15 minutes, or 68 mph. If we follow the triangle formed by these two velocities, the resultant vector is the hypotenuse of a triangle with sides of 30 and 68 mph, which is at an angle equal to the arctangent of 68/30 or about 66 degrees.

(2a) If we follow the triangle formed by the distances traveled, the resultant vector (displacement) is the hypotenuse of a triangle with sides of 10 and 17 miles, or about 20 miles. The direction is the arctangent of 17/10, or about 60 degrees.

(2b) If we follow the triangle formed by the travel times, the resultant vector (displacement) is the hypotenuse of a triangle with sides of 20 and 15 minutes, which is about 28 minutes. The direction is the arctangent of 20/15, or about 53 degrees.

(2c) Then the displacement velocity is 20 miles in 28 minutes or about 43 mph. For the direction, we would have to pick either the one from (2a) or (2b).

What is the answer? And why? Is it mere convention? If so, then we’re dealing with a symmetry.