ENG1005 S2 2022 Assignment 3: Designing a bobsled track

27 marks total

This assessment is intended for you to apply the mathematical skills you are learning in weeks

5-8 of the unit. It is also designed to practice communicating your work clearly.

General assignment information:

1. Electronic submission of this assignment is due on Moodle by 11.55pm on Wednesday 5

October 2022 (Melbourne time).

2. This assignment is not easy and will not be doable if you leave it to the last

minute. You have covered everything you need for this assignment after week 8.

3. Your assignment submission should include a description/explanation of what

you are doing at each step and relevant working. Without these you will receive

limited marks. The description should be in complete English sentences. All mathematics

should be appropriately laid out and with appropriate notation. Your writing should be

similar in style to the worked solutions from the problem sheets, not the annotations from

the videos. For more information and advice, please read the “Guidelines for writing in

mathematics” document posted under the “Assessment” tab on Moodle.

4. Your assignment may be typed or handwritten and scanned. The final document should

be submitted as a single pdf file that is clearly and easily legible. If the marker is

unable to read it (or any part of it) you may lose marks.

Getting help:

1. You are allowed to discuss the questions with your fellow students and even share ideas

about how to solve them. However, you are not allowed to show your written work to

anyone and you are not allowed to look at or copy the written work of anyone else.

2. Come to office/consultation hours. Please see the “Communication” section of Moodle for

time and location details.

3. In the Applied Classes, feel free to ask your instructor in the last 10 minutes of class. They

are savvy enough to provide assistance on the problems without giving too much away.

4. At Clayton, the Mathematics Learning Centre (MLC) is open 10 AM to 2 PM every weekday.

Academic integrity:

Please avoid the temptation to use any person or entity’s help outside of those listed. Asking

your personal tutor to do your assignment is considered cheating. Posting your assignment on

a “homework” website is considered cheating (even if only to check your answer). Making use

of solutions posted on such pages is considered cheating. And so on. Please refer back to your

Academic Integrity module if you are in any doubt. Your integrity is an important part of

who you are. It is much more important than any grade you could receive.

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Bobsledding involves hurtling down a steep, windy, narrow track of ice in a sled, at higher speeds than

allowed on most highways. For a track to be both challenging and safe, it requires extremely careful design.

This assignment will explore some calculations relevant to the preliminary design of a track.

Bobsled tracks are typically around 1.5 km long with numerous bends (as shown in the image below of

the Sanki track). The steepest grade is around 15 to 20%. Paths for the track are chosen so that the natural

slopes are used as much as possible.

By Sergei Kazantsev, CC BY-SA 3.0 https://creativecommons.org/licenses/by-sa/3.0 , via Wikimedia Commons

Part A: Track slopes

A track is planned for a ridge whose topography can be approximated by

h(x, y) = 1 + 0.1e

−3x

2+2xy−3y

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where h is the height above sea level in kilometres,1 x is the distance east of the peak in kilometres and y is

the distance north of the peak in kilometres.

You may find it useful to visualise this using Matlab with the following commands2

x = [-1:0.05:1];

y = [-1:0.05:1];

[xx,yy] = meshgrid(x,y);

h = 1 + 0.1*exp(-3*xx.^2 + 2*xx.*yy - 3*yy.^2);

mesh(xx,yy,h)

to get a surface plot, or replace the last line by

contour(xx,yy,h)

axis equal

to get a contour plot.

1. Find the gradient of h(x, y). [2 marks]

2. Hence find the magnitude of the steepest slope S(x, y) at each point (x, y). [1 mark]

3. Briefly describe how you would find the location of the overall steepest slope. (You do not need to do

any calculations.) [2 mark]

1Although metres would be more standard, we’ll use kilometres to keep the numbers around one.

2For those who have not used Matlab for this sort of plot before, note that the “.”s in line 4 are important. Without these,

Matlab will try to do matrix multiplications rather than point by point multiplications.

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4. Possible locations for the overall steepest slope are (0, 0), (1/4, −1/4), (−1/4, 1/4), (1/

√

8, 1/

√

8) and

(−1/

√

8, −1/

√

8). Using the contour plot of h(x, y), explain which of these points has/have the steepest

slope. [1 mark]

A proposed section of the track is given in parametric form by

x(ξ) = 0.1 sin(10πξ) − ξ and y(ξ) = 0.1 sin(10πξ) + ξ,

with z(ξ) = h(x(ξ), y(ξ)) and parameter values 0 ≤ ξ ≤ 1.

5. Using Matlab or other software, plot this curve in the (x, y) plane (in other words, plot the path of the

track viewed directly from above). [1 mark]

6. Find an expression for the tangent vector to this proposed track in the (x, y) plane (in other words

only considering the x and y components). [2 marks]

7. Use this tangent vector to find the slope along the track s(ξ). [3 marks]

8. Plot the height of the track z(ξ) and slope of the track s(ξ) that you have calculated against ξ, the

horizontal distance of the centre-line of the path from the peak. Be sure to include axis labels and

units on your plot. [3 marks]

9. If the steepest allowed slope is −0.2, are there any points where the slope is too steep on this track?

[1 mark]

Part B: Bobsled speeds

It is also important to explore the possible speeds that a bobsled could achieve on the track.

A simple but reasonably realistic model for how the speed V changes in time t is3

M

dV

dt = Mg sin(α) − µFn −

1

2

ρCDAV 2

.

This ODE represents Newton’s law: the left-hand-side is the mass M of the bobsled times its acceleration

and the right-hand-side is the sum of all the forces acting on it. The first term on the right is the force of

gravity along the slope with g the acceleration due to gravity and α the angle corresponding to the slope.

The second term is the frictional force given by a coefficient of friction µ times the normal force Fn. The

third term is air drag on the bobsled where ρ is the density of air, CD is a drag constant and A is the effective

cross-sectional area of the bobsled.

Realistic values for the parameters are M = 600 kg, g = 10 m s−1

, µ = 0.01, ρ = 1 kg m−3

, CD = 0.3,

A = 0.4 m2

.

This model is valid for curved tracks with varying slopes, however it cannot be solved by hand in this

case. Therefore for the assignment we will assume the track is straight with a constant slope. In this case

Fn = Mg cos(α) and we will assume α ≈ 6.3

◦

so that g sin(α) − µg cos(α) = 1.

With these parameter values, the ODE becomes

dV

dt = 1 − 10−4V

2

.

1. Solve this ODE to find V (t) subject to the initial condition V (0) = 0. [3 marks]

2. What is the speed at long times if the ramp is arbitrarily long? [1 mark]

3See Heck A, Uylings P. Gliding for Olympic success. Physics Education. 2022 Mar 15;57(3):035011 for more details.

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3. The distance D(t) that the bobsled has travelled along the track from the start satisfies the ODE

dD

dt = V.

Integrate this equation to find D(t). [2 marks]

4. Plot the speed against position for a track of length 1500 m. Again be sure to include axis labels and

units on your plot. Hint: time can be thought of as a parameter for this plot. [1 mark]

There are also 2 additional marks given for the quality of the presentation and 2 additional

marks for correct mathematical notation.

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