Answer
$s(t)=475t-25t^{2}$
Work Step by Step
We equate speed and velocity as there will be no change in direction of motion.
Subtracting the head wind speed to the speed of $500$ mph,
$v(t)=500-(25+50t)=475-50t$
Velocity is the derivative of position, so position is an antiderivative of velocity.
$s(t)=\displaystyle \int(475-50t)dt=475t-50\cdot\frac{t^{2}}{2}$
$s(t)=475t-25t^{2}$
(There is no constant C because $s(0)=0$.)
