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Category  Math 

Subject  Other 
Difficulty  Undergraduate 
Status  Solved 
More Info  University Math Homework 
Short Assignment Requirements
Assignment Description
Coursework Specification 1
Aim & Rationale
Aim
The aim of this assignment is to use MATLAB to create a mathematical model to predict the top speed of a gravity racer.
Why are we asking you to do this?
The skills you demonstrate in this assessment are important because mathematical modelling, using programmes such as MATLAB, is a widely used method for design and analysis in all engineering disciplines. Being able to concisely and coherently present your work in a report is an important method of communication for all engineers.
Task Description
For this assessment you are required to use MATLAB to create a mathematical model of a gravity racer travelling down an inclined slope. The model should be able to predict the speed of the racer at any given distance.
Report
PDF Document. Max 10 sides A4, including Appendix
Report
Report Content
You are required to submit an individual report for this assessment.
The report is to include a flowchart of your solution, your MATLAB code as well as the answers to the questions in the assignment brief. Please follow the guidelines below for advice on report structure and style.
Report Structure
Critical Analysis
Report Checklist
Style Guides
Report Structure and Style
Follow the guidance on writing reports published in the Student Toolkit on the School of Engineering Students Moodle area.
Your report should have the following sections:
Title
Introduction
Methods
Results
Discussion
Conclusion
Acknowledgements
References
Appendices
Assignment Description
Unit  Assignment 
 
Code:  6E4Z1104  ID:  1CWK60 
Title:  Mathematical Methods 1  Title:  Assignment 1: Predictive model of a gravity racer 
Leader:  Heather Driscoll  Coordinator:  Heather Driscoll 
For students at MMU, studying with/without attendance 

Task Details & Instructions
For this assignment, you are required to create a mathematical model to predict the speed of a gravitypowered vehicle.
Gravity racers are one of the simplest designs of a vehicle relying only on gravitational forces to derive forward motion. Known as soapboxes or downhill gokarts, the vehicle class of gravity racers consist of a fourwheeled chassis, together with a steering and braking system. Although simple in concept, the design of a gravity racer requires an understanding of a number of engineering and physical science topics. In order to optimise the design of a racer to achieve the fastest possible speed, you have been asked to create a predictive model. The predictive model should be a computational solution to the equations describing the motion of the racer. Using values of known variables such as the frontal area and mass of the racer together with information on the racetrack, such as slope angle, the predictive model should be used to solve Newton’s second law and calculate the acceleration of the racer. The velocity and displacement of the racer should then be found from the derivatives of the acceleration.
These calculations should be solved iteratively using the Forward Euler approach in MATLAB.
Fig. 1 Example of a gravity racer used to set the Guinness World Record in 2014.
Task Deliverables
Write a short report comprising of the sections below. Follow the guidelines on Moodle for correct report structure. MATLAB must be used for this assignment. Save your report as a PDF document and follow the submission guidelines on Moodle.
1. INTRODUCTION
• Briefly introduce the problem and the rationale for undertaking the work.
• You do not need to include a literature review.
2. METHOD
The method section should include the following:
• Flow chart illustrating how you have designed your code MATLAB code including comments
3. RESULTS
Use your MATLAB code to answer the following questions:
Q1. Using the standard variables, what is the predicted speed of the racer (in ms^{1} and mph) at 2000 m in the following scenarios?
a) No air resistance and no rolling resistance
b) Air resistance included, but no rolling resistance
c) Both air resistance and rolling resistance included
For all scenarios, include a graph of speed against distance.
Standard variables for Q1  
Mass of the racer, m  200 kg 
Drag coefficient, C_{D}  0.26 
Frontal area, A  0.42 m^{2} 
Coefficient of rolling resistance, _{R}  0.004 
Slope angle,  4 
Q2. The racer has a number of components that can be altered or changed. From the options provided, select the optimum combination of components that produce the highest speed at 2000 m for the standard slope angle of 4°. State your chosen components and the predicted speed.
Component selection for Q2 – select one Frame and one set of Wheels  
Frame  Steel chassis m = 216 kg A = 0.5 m^{2} C_{D} = 0.31  Carbon fibre monocoque m = 160 kg A = 0.36 m^{2} C_{D} = 0.22 
Wheels  Disc wheels m = + 2 kg* C_{D} =  0.02* _{R} = 0.006  Spoke wheels m =  1 kg* C_{D} = + 0.01* _{R} = 0.003 
* To be added or subtracted from the mass and drag coefficient values of the chosen frame.
Q3. Not all racetracks consist of a constant slope angle. Modify the code to predict the top speed of the racer at racetrack B using the standard variables for the racer.
Slope angle profile at racetrack B for Q3  
0 distance 500 m  = 5 
500 distance 1250 m  = 2 
1250 distance 2000 m  = 7 
2000 distance 3000 m  = 3 
4. DISCUSSION
• Comment on the difference in magnitude the effect of air resistance and rolling resistance have on the predicted speed of the racer. To help your discussion, include a graph with all three results from Q1 presented.
• Comment on the racer configuration selected in Q2. Why do you think that selection of components are able to generate the highest predicted speed?
• At what distance along the track did the racer reach top speed in Q3? Why might this be important when selecting a venue to set a new world record?
5. CONCLUSION
Summarise your findings and discuss the success of your model.
Support
Members of the unit team will discuss individual assignment queries during tutorials, lab sessions or in office hours. Before going to see a member of the unit team make sure to try the following:
 Complete any lab worksheets, tutorial questions or formative assessments, particularly those on subjects related to the coursework and ensure that you understand the relevant topics.
 If you are struggling with formative work or understanding theory, please ask for help during tutorials or lab sessions before seeking coursework advice. Remember that the unit team and tutors can give you far more detailed help with formative tasks such as lab worksheets than they can with your assessments including worked solutions and/or answers.
If you need support outside of unit team office hours, in particular on general skills such as report writing or planning your coursework, please refer to the Moodle information or contact the Programme Support Tutors.
Programme Support Tutors  E323, ..., tel: 0161 247 1600 
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PLPrelated advice  Dr Prasad Ponnapalli, ... 
Understanding the physics of the problem
There are three main forces acting on the gravity racer when travelling down an inclined slope (Fig.2):
(1) Force due to gravity, F_{W}
(2) Air resistance (drag), F_{D}
(3) Rolling resistance, F_{R}
Using Newton’s second law (F = ma) we can calculate the acceleration, a, of the racer by resolving the forces:
F_{A} = ma = F_{W} – F_{D} – F_{R} 
 (Eq.1)  Fig. 2  Freebody diagram. 
The individual force components can be calculated from the following equations:
F_{W} = m g sinθ F_{D} = ½ ρ v^{2 }C_{D }A F_{R} = m g μ_{R }cosθ


 (Eq.2) (Eq.3) (Eq.4)  m = mass of the racer in kg g = acceleration due to gravity, 9.81 ms^{2} θ = slope angle in degrees ρ= air density = 1.225 kgm^{3} v = velocity of the racer in ms^{1} C_{D} = drag coefficient A = frontal area of the racer in m^{2} μ_{R} = coefficient of rolling resistance 
The solution to Eq.1 allows the speed of the racer to be predicted at any given point on a potential racetrack. This is because the velocity of the racer is the derivative of the displacement with respect to time, and the acceleration of the racer is the derivative of the velocity with respect to time.
The Forward Euler method can be used iteratively to find the velocity and displacement of the racer given the acceleration:
x_{n+1 }= x_{n} + Δt v_{n} (Eq.5) v_{n+1 }= v_{n }+ Δt a_{n} (Eq.6)
where x_{n} is the displacement at time t and x_{n+1} is the displacement at time t + Δt; similarly for velocity, v and acceleration, a.
Further reading
For more information on the physics of the problem, please refer to following journal paper (a link is provided on Moodle):
Driscoll, H.F., Bullas, A.M., King, C.E., Senior, T., Haake, S.J. and Hart, J. (2016) ‘Application of Newtonian Physics to predict the speed of a gravity racer.’ Physics Education, 51(4).
Marking scheme
 Mathematical Methods 1 
 
 Documentation 25%  Methods 25%  Results 25%  Discussion 25% 
5  Professionally presented and wellstructured report. Numbers and appropriate captions on figures and tables. Appropriate scientific writing style (3^{rd} person). Free from grammatical errors and spelling mistakes. Introduction demonstrates strong understanding of the problem and mathematics required. Conclusion includes a clear, concise and accurate summary of the report. Clear and easy to follow.  Flow chart is clearly presented and shows an appropriate and logical sequence of procedures, including inputs, calculations and outputs. MATLAB code is well structured and annotated. The code produces a correct solution and is efficient to run with suitable timestep chosen.  Results are clearly stated and all questions are correct. Where required, graphs are presented with appropriate labels and titles. Answers are given to an appropriate number of significant figures and correct units.  Clear and thorough discussion of results addressing all the points on the assignment brief. Solutions from Q1 presented on the same graph with appropriate labels and title. Strong understanding of engineering influence on results. 
4  Wellpresented and structured report. Figures are numbered with appropriate captions. Scientific writing style. Mainly free from grammatical errors and spelling mistakes. Introduction demonstrates a good understanding of the problem and mathematics required. Conclusion includes a concise summary of the report.  Flow chart is well presented and shows an appropriate sequence of procedures, including inputs, calculations and outputs. MATLAB code is well structured with some annotation. The code produces a correct solution with suitable timestep.  Results are clearly stated but some errors in the solutions. Where required, graphs are presented with appropriate labels and titles. Answers given to an appropriate number of significant figures and correct units.  Discussion of results addressing all the points on the assignment brief. Solutions from Q1 presented on the same graph with appropriate labels and title. Demonstration of understanding of engineering influence on results. 
3  Structured report. Some figures numbered with captions. Scientific writing style but not always written in 3^{rd} person. Some grammatical errors and spelling mistakes. Introduction demonstrated an understanding of the problem and mathematics required. Conclusion includes a summary of the report.  Flow chart is adequately presented but is lacking in detail on the sequence of procedures. MATLAB code is sufficient to produce the correct solution but limited annotation or thought on the efficiency to run.  Not all questions have been attempted or there are significant errors in the solutions. Graphs are presented but missing labels or titles. Answers given to inappropriate number of significant figures or erroneous units stated.  Discussion of results addressing some of the points on the assignment brief. Solutions from Q1 presented with some errors. Some understanding on engineering influence on results. 
2  Report lacks structure or is incomplete. Figures are unclear and are missing numbers and captions. Number of grammatical errors and spelling mistakes. Introduction demonstrated limited understanding of the problem and mathematics required. Conclusion missing key information.  Flow chart is missing key procedures or demonstrates an illogical sequence. MATLAB code is presented but is not suitable to solve the problem.  Limited results presented with significant errors in the solutions. Graphs are poorly presented or missing.  Limited discussion of results. Solutions from Q1 not presented or with significant errors. Little understanding of engineering influence on results. 
1  Poorly structured and poorly presented. Sections are missing or severely lacking in detail. Introduction shows little or no understanding of problem or mathematics required. Missing or incorrect conclusion. Poorly written and hard to read/follow.  Flow chart is erroneous or extremely limited in detail. MATLAB code is incomplete and insufficient to solve the problem.  Results are incorrect and poorly presented.  Poor or incorrect discussion of results. Solutions from Q1 not presented or with significant errors. Incorrect or no understanding of engineering influence on results. 
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