Assume the resistive force acting on a speed skater is proportional to the square of the skater's speed $v$ and is given by $f=-k m v^{2},$ where $k$ is a constant and $m$ is the skater's mass. The skater crosses the finish line of a straight-line race with speed $v_{i}$ and then slows down by coasting on his skates. Show that the skater's speed at any time $t$ after crossing the finish line is $v(t)=v_{i} /\left(1+k t v_{i}\right) .$