Q4. Consider an advanced model which describes a car competing in a drag race. Assume that the car begins from stationary at the start line and accelerates at full power towards the finish line.

(a) The relationship between the power of the engine and the acceleration created by the engine is given by: Power=Mass x Acceleration x Velocity

i. Write the relationship in terms of a differential equation which models the velocity of the car (generated only by the engine) and find a particular solution for the velocity of the car at any time using the initial data from the race.

ii. How long does it take for a car with a maximum power of 300,000 Watts and a mass of 1,500Kg to make a drag race (400 meters), only taking into account the power of the engine and the initial race data?

(b) Moreover, a differential equation which describes the velocity of a car in a drag race and takes air resistance into account is given by: v'(t)=a(t) - k[v(t)]^2 where a(t) is acceleration generated by the engine (can be derived from power-acceleration relationship) and k>0 is the drag coefficient.

i. Consider a car with drag coefficient k=1/7500, mass 1,500kg and a maximum power of 300,000 Watts. Use Euler's method at 10 equidistant points in time for t is element of [1,10] to approximate the velocity of the car given that v(1)=4.

ii. Plot the corresponding individual approximations for velocity and acceleration derived using Euler's method on a velocity-time graph and an acceleration-time graph.