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BTEC Unit 7 Calculus Assignment: Velocity Model, Position & Acceleration Analysis for Racing Car Performance
| University | Milton Keynes College |
| Subject | Unit 7: Calculus to solve engineering problems |
Unit number and title
Unit 7: Calculus to solve engineering problems
Learning aim(s)
C: Investigate the application of calculus to solve a defined specialist engineering problem
Assignment title
Investigate an engineering problem using calculus
Vocational Scenario or Context
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_max))
Where:
- v(t) is the instantaneous velocity of the car (m/s)
- t is the time in seconds
- t_max 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.
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Task 1
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:
- Use thinking methods to analyse the given engineering problem, e.g. break the problem down into a series of manageable elements, and produce a specification
- Prepare a valid proposal for solving the problem and present it
- Produce mathematical models for the identified elements of the problem
- Apply calculus methods to produce answers for each of the elements
- Bring the elements together in a formal presentation
In your proposal you need to outline the problem and the methods needed to solve it. You need to include how to:
1. Model Analysis
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. Velocity Graph
Sketch a graph of velocity vs. time.
3. Position Function
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. Position Graph
Sketch a graph of position vs. time.
5. Acceleration Function
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. Acceleration Graph
Sketch a graph of acceleration vs. time.
7. Application to Specific Car
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
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Checklist of evidence required
Your informal report should contain:
- written descriptions
- analysis
- worked solutions to the problems
Each worked solution should be laid out clearly and contain brief explanations of the stages of the calculation to indicate your understanding of how calculus can be used to solve an engineering problem. Diagrams and sketches should be well presented and clearly labelled.
Criteria covered by this task:
7/C.D3 Critically analyse, using technically correct language and a logical structure, a complex engineering problem, synthesising and applying calculus and a mathematical model to generate an accurate solution.
7/C.M3 Analyse an engineering problem, explaining the reasons for each element of the proposed solution
7/C.M4 Solve accurately, using calculus methods and a mathematical model, a given engineering problem.
7/C.P7 Define a given engineering problem and present a proposal to solve it
7/C.P8 Solve, using calculus methods and a mathematical model, a given engineering problem
Sources of information to support you with this Assignment
- mathsisfun.com/index.htm
- mathcentre.ac.uk/students/topics
Other assessment materials attached to this Assignment Brief
| Candidate | Car model | t(0-28 m/s) (s) | t(400m) (s) | tmaxspeed (s) |
|---|---|---|---|---|
| 1 | 1994 Ford SVT Boss Mustang 10.0L Concept | 1.9 | 10.50 | 7.1 |
| 2 | 2002 Chevrolet Corvette (Lingenfelter 427 Twin Turbo) | 2.0 | 9.10 | 6.8 |
| 3 | Bugatti Veyron 16.4 Grand Sport Vitesse – [2012] | 2.5 | 9.21 | 10.2 |
| 4 | Caterham 7 620 R 2.0 L Supercharged – [2013] | 2.8 | 10.91 | 7.0 |
| 5 | Ariel Atom V8 500 – [2008] | 2.3 | 10.80 | 5.0 |
| 6 | Bugatti Veyron 8.0 litre W16 Super Sport – [2010] | 2.5 | 9.75 | 8.0 |
| 7 | Caparo T1 2.4 V8 – [2006] | 2.5 | 8.23 | 7.0 |
| 8 | Hennessey Venom GT – [2010] | 2.5 | 8.64 | 8.5 |
| 9 | Radical RXC 3.5 V6 Twin Turbo – [2014] | 2.6 | 10.46 | 7.0 |
| 10 | Ascari A10 5.0 V8 – [2006] | 2.8 | 10.36 | 7.8 |
| 11 | Lamborghini Veneno 6.5 V12 – [2013] | 2.8 | 9.92 | 8.5 |
| 12 | Porsche 918 Spyder 4.6 V8 – [2013] | 2.8 | 11.49 | 8.2 |
| 13 | Caterham 7 Superlight R500 – [2009] | 2.9 | 11.32 | 8.0 |
| 14 | Ferrari F12 tdf 6.3 V12 – [2015] | 2.9 | 10.18 | 7.4 |
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