A radio controlled unmanned boat is propelled by a 20 kilowatt (kW) electric motor (1 kW = 1000 watts). Fully loaded boat weighs 400 kilograms (kg) and experiences the resistance force, in Newtons, proportional to the boat's speed, in meters per second, with proportionality constant k=50 N.sec/m.
Note that the power P of an engine in watts (W) and its locomotive force Fₑ in Newtons (N) are related by P=Fₑv, where v is the velocity in meters per second (m/sec).
(a) Set up a mathematical model describing the motion of the boat accelerating from rest with motor engaged at full power. Your model must include a diagram, a consistent system of units, the variables used to describe this system, and the differential equation for the velocity v of the boat as a function of time along with the initial condition. Use Newton's second law F=ma.
(b) Use this model to determine the limiting velocity of the boat and the time it would take for the boat to reach the speed of 54 kilometers per hour (km/h) (1 m/sec = 3.6 km/h). Leave the answer in an "algebraic" form, which may involve fractions, radicals, and logarithms. Also, remember to convert all quantities into your system of units and include appropriate units in your answers.