Putnam 1946 A6

A particle of unit mass moves on a straight line under the action of a force which is a function $f(v)$ of the velocity $v$ of the particle, but the form of the function is not known. A motion is observed, and the distance $x$ covered in time $t$ satisfies the formula $x= at^2 + bt+c$ , where $a,b,c$ have numerical values determined by observation of the motion. Find the function $f(v)$ for the range of $v$ covered by the experiment.