Q: A canoer crosses three-fifths the distance across a lake at speed ( v ). Unfortunately, the canoer begins to tire, and she finishes crossing the lake at one-half her initial speed. What was her average speed while crossing the lake?
A: The average speed is determined by dividing the total distance traveled by the total time of travel; however, the rate of travel changed after three-fifths the distance was traveled. Thus, we must first determine how much time transpired as these distances were crossed:
( d1 / v1 ) = t1
( d2 / v2 ) = t2
( 0.6 d / v1 ) + [ 0.4 d /( ½ v1 ) ] = ttotal
It is no longer necessary to use the ( v1 ) term, so…
( 0.6 d / v ) + ( 0.8 d / v ) = ( 1.4 d / v ) = ttotal
dtotal = ( 0.6 d ) + ( 0.4 d ) = d
( d / ttotal ) = [ d / ( 1.4 d / v ) ] = ( v / 1.4 ) = 0.71 v
RENAISSANCE PHYSICS 