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MATHEMATICAL METHODS FOR ENGINEERS 1 (ENR 114)

Group Project, SP3 2020

See unit website for due date

Ground Rules for Students

- This is a group project, and each group has exactly two students.
- The two students must share the work equally between them.
- You have limited time to find a partner. If you do, inform the course coordinator by email or by a message through the web site. Students who do not find a partner will be

randomly assigned to a group. - Students may be given permission to do it individually. However, if a student elects to do

the project individually, then that student takes full responsibility for the extra workload.

The same applies to students who, through their own actions, end up doing the project on

their own. - Collaboration between partners will take place online, through email, OneDrive, and Lync

(a Virtual Classroom tool). - Troubleshooting. It sometimes happens that one partner feels that they are carrying

the group, and that the second partner is not contributing. When that happens, the unit

coordinator needs to be informed immediately. - Troubleshooting. It sometimes happens that two partners do not get along, even though

both are contributing to the project. If you feel uncomfortable in your group, inform the

unit coordinator as soon as possible. - Most importantly, one report (including mfile) per group must be submitted. The front

page of the project must have the name of both partners.

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A Bouncing Ball

Consider a ball allowed to fall onto a hard surface and rebound, continuing to bounce until it is

at rest. In the analysis undertaken here, the ball is dropped, i.e. its initial velocity is zero, and

the surface it lands on is perpendicular to its flight. A ball with high resilience, for example,

a new table tennis ball, retains most of its kinetic energy following the collision of ball and

surface, and it rebounds to a high proportion of its original height.

A simple model of the ball’s behaviour views the movement of the ball in one-dimension, and

would ignore air resistance. There is loss of energy in the collision; some energy is transformed

into sound, some into a slight rise in the heat of the ball following the deformation and compression it experiences during the impact, and some energy may impart a spin to the ball. The

dissipation of energy can be accumulated into the term resilience, a characteristic of various

types of balls. The main factors governing the height of the ball’s subsequent bounce are; its

initial height (and thus velocity), the force of gravity, and the ball’s resilience. One measure of

resilience is called the coefficient of restitution, commonly given the symbol e.

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Task 1

We assume that the potential energy of the ball at its initial position ( P.E. = mgh , where

m is the mass of the ball, g is the gravitational constant (take g = 9.81 m/s2 here), and h is

the height of the ball) is entirely converted to kinetic energy (K.E. =

1

2

mv2

, where v is the

velocity of the ball just before it hits the surface).

Note: it may be easier to take the positive direction as being downward for this problem.

a) Show that the ball’s velocity just before it hits the surface is v =

√

2gh .

Denote the maximum height of the ball before each bounce with a subscript matching the

number of the following bounce. Let the velocity of the ball just before its first bounce be

v1 =

√

2gh1 , then its velocity just before its second bounce is v2 =

√

2gh2 . By symmetry,

this has the same magnitude as the ball’s velocity just after its first bounce, and the coefficient

of restitution is calculated as the ratio of these, i.e. e =

|v2|

|v1|

. Note that 0 < e < 1 where

e = 1 would be perfect elasticity.

b) Show that e =

r

h2

h1

.

Since we have constant acceleration while the ball either falls or bounces upward, the time

for the ball to fall from its initial position to impact with the surface can be calculated as

∆t =

vf − vi

a

, where vf is velocity at the end of the time interval, vi

is velocity at the start

of the time interval, and a is the acceleration applied.

c) Show that the time for the ball to fall from its initial position until impact is t1 =

r

2h1

g

.

d) If t2 is the time for the ball to fall from (or to reach) its subsequent maximum rebound height,

show that t2 = et1 .

Note that the flight time between the first and second bounces is 2t2 .

e) Generalise the recurrence relation in part d) to show that the time until the ball comes to rest

is

T = t1 + 2t2 + 2t3 + … = t1 + 2et1

X∞

i=0

e

i =

s

2h1

g

1 + e

1 − e

.

This equation for T says that although the model allows for an infinite number of bounces,

they occur within finite time. And that time until the ball comes to rest depends on the square

root of the ball’s initial height, which seems reasonable.

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Task 2

a) Demonstrate that the time when the ball attains maximum rebound height is

t(hi) = X

i−1

j=1

tj (1 + e)

for t(h2), t(h3) and t(h4).

I do not have experimental data to provide so the following plot only illustrates a theoretical

model. A competition-approved table tennis ball (mass 2.7g, diameter 40mm) must have a

coefficient of restitution (using a steel block as the surface) of 0.89 to 0.92, use a value of

e = 0.9 here. Take initial height as 0.5 m.

b) Construct a for loop in MATLAB to calculate the maximum rebound heights and the time

they occur until the ball is at rest. Use the scatter command to plot this data using circles

to indicate the position of the ball. Have the colour of the circles grade from yellow to red

across the plot. Comment on the features of the graph.

c) Does the time to rest calculated using the equation in part e)of Task 1, and estimated from

the graph of part b) above agree? Comment on this.

Task 3

The relationship ti = eti−1 says that the duration of a rebound is proportional to the duration

of the preceding rebound. This implies an exponential relationship (in this case, exponential

decay) between successive rebound durations.

a) Calculate the natural log of rebound durations (t1, t2, …) and produce a scatter plot in MATLAB of these values plotted against bounce number. Fit a line (use MATLAB ) to the plotted

points and display the equation of the line on the plot. Comment on the result.

b) Calculate the time between successive bounces and the time when the first of these bounces

takes place. Produce a scatter plot of these data in MATLAB . Fit a line (use MATLAB )

to the plotted points and display the equation of the line on the plot. The slope of this line

should be e − 1.

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