You are working as a Junior Engineer for a small motor racing team. You have been given a proposed mathematical model to calculate the velocity of a car accelerating from rest in a straight line. The equation is: v(t)= A(1-e^-t/tmax) v is the instantaneous velocity of the car (m/s) t is the time in seconds Tmax is the time to reach the maximum speed in seconds A is a constant. However, the time taken to reach the steady state value (maximum speed) is approximately 5 time constants, so the model as it stands cannot work accurately. A modified model recognising this issue would be to change it to: v=A(1-e^-5t/tmax) The Team Manager has asked you to carry out an analysis on the models and produce a written report. As part of the analysis you have been given some data on a known model. Produce a report that contains written descriptions, analysis and mathematics that shows how calculus can be used to solve an engineering problem. The tasks are to: Identify the 1. units of the coefficient A physical meaning of A velocity of the car at t = 0 asymptote of this function as t → [infinity]? 2. Sketch a graph of velocity vs. time. 3. Derive an equation x(t) for the instantaneous position of the car as a function of time. Identify the value x when t = 0 s asymptote of this function as t → [infinity] 4. Sketch a graph of position vs. time. 5. Derive an equation for the instantaneous acceleration of the car as a function of time. Identify the acceleration of the car at t = 0 s asymptote of this function as t → [infinity] 6. 6. Sketch a graph of acceleration vs. time. 7. Apply your mathematical models to your allocated car. Use the given data for the 0 – 28 m/s and 400m times to calculate the: value of the coefficient A maximum velocity maximum acceleration. t(0-28m/s) (S) 1.5 t(400m) 10.5 sec tmax 7.1