Given that the exact solution to (?) is:v(t) = 50e−14t,determine whether the numerical solution found in (a) is accurateto within a5%relative errorat each time step.

A car is being tested around a racetrack and, during a lap, the speed ofthe car is recorded every 10 seconds:Time (s)010 20 30 40 506070 80 90 100Speed (ms−1) 90 66 73 58 67 86 102 85 79 90 106Using the table:

(a) Create a point plot of the velocity data.

(b) Approximate the acceleration of the car at each velocity measurementby numerical differentiation using:ˆaforward differenceapproximation at timet= 0s;ˆabackward differenceapproximation at timet= 100s; andˆacentred differenceapproximation otherwise.

(c) Create a point plot of the approximated acceleration data.

(d) Approximate the length the car travelled during these measurementsusing numerical integration using:(i) the trapezium rule(ii) Simpson’s rule.
School of EngineeringBEng Automotive Performance Engineering Portfolio Exercise I: 2020–21MSP5017 Engineering Mathematics IIDUE: 5pm,18thDecember 2020(15 marks)(6 marks)(4 marks)

QUESTION 2

As a car moves there is a force resisting the movement that is proportional to the car’s velocityv=v(t). If the car has mass mthen from Newton’s Second Law:ma=F=⇒m ̇v=−κv=⇒ ̇v=−κmvwhere ̇v=dvdtandκis a measure of the resistance to the motion dueto friction and air resistance. Assume we have a car that weighs1000kgandκ= 250kgs−1, then if the car begins tocoastatt= 0with speedv(0) = 50ms−1, we have the first-order differential system:(?) ̇v=−14vv(0) = 50

(a) Use Euler’s method with a step size ofh= 0.4sto solve (?) anddetermine how long it takes for the velocity of the car to halve.

(b) Given that the exact solution to (?) is:v(t) = 50e−14t,determine whether the numerical solution found in (a) is accurateto within a5%relative errorat each time step.

(c) Create a point plot of your numerical results and superimpose a plotof the analytic solutio