$4\mathrm{.}28$ A sled used to test rockets is slowed lowering a scoop into a water trough. The scoop is $0\mathrm{.}3$ m wide and deflects water through $150.$ The mass of the sled is $8000$ kg $.$ At the initial speed, it experiences an aerodynamic drag force of $90$ kN $.$ The aerodynamic force is proportional to the square of the sled speed. It is desired to slow the sled to $100\frac{m}{s}.$ Determine the depth the scoop must be lowered into the water to slow the sled down from an initial speed of $300\frac{m}{s}$ to $100\frac{m}{s}$ in $800$ m $.($Note that using the chain rule the derivative can be expressed as $\frac{dU}{dt}=\frac{dU}{dx}\frac{dx}{dt}=\frac{dU}{dx}U)$