You need to go from point A to point B, and then back to point A. Points A and B are 20 miles apart. You go to point B at a constant speed of 15 miles per hour. If you want your overall speed to be 30 miles per hour, how fast would you have to go from point B back to point A?

(In reply to solution by amanda)

Unfortunately, speeds do not average linearly over distance.  Traveling distances D1 and D2 at speeds R1 and R2, takes times T1=D1/R1 and T2=D2/R2, resp. The average speed is thus (D1+D2)/(T1+T2)= 1/(W1/R1+W2/R2) where W1=D1/(D1+D2) and W2=D2/(D1+D2). For this problem W1=W2=1/2 so the average speed is 1/(1/2R1+1/2R2) which is known as the "harmonic mean"  of R1 and R2 -- it is the reciprocal of the ordinary mean of the reciprocals. Thus with R1=15 and R2=45, the average speed would be 1/(1/30+1/90)=1/(4/90)=22.5. It is clear that 1/(1/2R1+1/2R2) < 1/(1/2R1)=2R1 -- to get the average speed to equal 2R1 would require R2="infinity."

Edited on March 12, 2004, 4:30 pm

Posted by Richard on 2004-03-10 20:17:40