Q4E Case Study 3
 The Coefficient of Restitution
Proposed Subject usage:
Mathematics / Physics (A/AS level)
Sports Science (1st/2nd yr)

AQA Alevel Physics 


Newton’s law of restitution. Applications
to direct impact of two particles and normal impact of
a particle with a plane surface.
Knowledge of the terms perfectly elastic
(1 = e) and perfectly inelastic/plastic (0 = e) is expected. 

Coefficient of Restitution
Direct impacts of two objects occur when their
velocity vectors are parallel. Linear momentum is conserved
but energy is lost during the impact when one or both of the
objects are deformed. The greater the deformation that occurs
the longer the impact lasts. The strain energy stored during
the deformation will partly be regained in the latter part of
the impact depending on how plastic or elastic the impact is.
Newton’s empirical law demonstrates this:
u1 – u2 = e(v1 –v2)
Where:
u represents the approach velocity
v is the separation velocity
1 and 2 represent the two objects
e = a constant known as the coefficient of restitution
Method 1 Using velocities to determine
e
If the impact is between an object (e.g. a tennis
ball) and a surface then there is no velocity prior to or after
impact for object 2.
Therefore: as u2 = v2 = 0 in the above equation : v/u = e
Method 2  Using heights to determine
e
v2 = u2 + 2as (one equation of uniform motion) Where: a = gravity (g) s = distance (in this
case height)
Therefore: u = v2ghd :
v = v2ghb
Where: hd = height dropped (prior to impact)
hb = height bounced (after impact)
By substituting u = v2ghd and v = v2ghb into
v/u = e
e =v(hb / hd )
This law indicates how e and velocity before
impact affects movement after impact. The value of e can range
from 0 – 1 with 0 being a perfectly plastic impact and 1 being
a perfectly elastic impact.
Methods
Five different sports balls dropped onto a wooden
table and filmed at high speed (100fps Basler). The videos were
digitised and the data were exported into excel files. Heights
of the balls were determined using the distance measuring tool
within Quintic Biomechanics 9.03
(Quintic Lite, Sports and Coaching
users can calculate distance by using the lines function and
onscreen coordinates). Both methods of calculating e (using
bounce height and velocity) can be used and compared.
Functions of the Quintic
Software used:
 Digitisation module
 Calibration
 Export data
 Lines and distance measuring tool
 Split screen
 Onscreen point coordinates
Results
The results shown in the tables below indicate
that there is a distinct difference between the impacts of the
different balls onto the surface.

Method 1 
Table
Tennis
Ball

Marble

Cricket
Ball

Tennis
Ball

Golf
Ball


Velocity
prior to impact (u) (m/s) 
1.92 
1.55 
1.39 
2.09 
1.66 
Velocity
after impact (v) (m/s) 
1.8 
0.82 
0.6 
1.76 
1.23 
e 
0.94 
0.53 
0.43 
0.84 
0.74 

Method 1 
Table
Tennis
Ball

Marble

Cricket
Ball

Tennis
Ball

Golf
Ball


H
dropped (m) 
0.22 
0.16 
0.13 
0.27 
0.16 
H bounced
(m) 
0.18 
0.05 
0.03 
0.18 
0.1 
Hb/Hd 
0.82 
0.31 
0.23 
0.67 
0.63 
e 
0.90 
0.56 
0.48 
0.82 
0.79 
The coefficients of restitution vary according
to the ball used; the tennis ball had the highest value (e =
0.82) and the cricket ball the lowest value (e = 0.48). This
means the tennis ball bounced much higher than the cricket ball
because much less energy was lost during impact. Therefore the
tennis ball did not decelerate as much and rebounded with a
higher velocity.
Conclusion
Coaches can apply this study advantageously
to increase knowledge of their sport by comparing different
sports, balls and surfaces. Understanding of biomechanical factors
involved in sport will be beneficial to the athlete and can
be used to predict and enhance performance. Quintic Software
is ideal for capturing and analysing video footage of athletes
and applying biomechanical knowledge to training and coaching
strategies of both individuals and the club or team.
Teachers may use the Quintic Software to engage
students’ interest by applying the fundamental principles of
topics such as the coefficient of restitution to sporting situations,
using reallife video examples. For example, from this case
study it is evident that the coefficient of restitution is very
important in sport, as different balls will behave differently
in similar situations. Perhaps it would be interesting to question
the implications of using certain sports balls in other sports
or discuss the benefits of warming up a squash ball. Using Quintic
Software teachers can make the syllabus more interesting and
enable increased understanding of difficult topics by putting
them in an appealing sporting context.
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