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/t_maxspeed )

v(t) is the instantaneous velocity of the car(m / s)

t is the time in seconds

t_maxspeed is the time to reach the maximum speed in seconds

A is a constant.

The Team Manager has asked you to carry out an analysis of the model and produce a written report. As part of the analysis you have been given some data on a known model.

Data:

Car Model = Bugatti Veyron t(0-28m/s)(s) = 2.5

t(400m) = 9.75

t_maxspeed(s) = 8.0

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/t_maxspeed)

Produce a report that contains written descriptions, analysis and mathematics that shows how calculus can be used to solve an engineering problem. Your Answer must include :

• methods to analyse the given engineering problem

• a valid proposal for solving the problem and present it

• Apply calculus methods to produce answers for each of the elements

• Bring the elements together in a formal presentation(most important)

Your Question:

In your proposal you need to outline the problem and the methods needed to solve it. You need to include how to:

1. Identify the:

● units of the coefficient A

● physical meaning of A

● velocity of the car at t = 0

● asymptote of this function as t → ∞?

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 → ∞

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 → ∞

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.

Additional information: To achieve the starting marking criteria you must show that you are able to: Solve accurately, using calculus methods and a mathematical model, a given engineering problem

From Question 1 to 7

Personal Added Information: Please explain everything step by step as I have to present this with some other car model values, in terms of a presentation, I can use any notes, so please explain very clearly in each step you go.