Multiple bounces of different material balls in free fall
Armando C. Perez Guerrero Noyola and Fernando Yañez Barona.
Departamento de Física, Ciencias Básicas e Ingeniería. Universidad Autónoma
Metropolitana- Iztapalapa,
Abstract
In this work, we present a proposal for a more dynamical experimental physics course for
physics and engineering students of the Universidad Autónoma Metropolitana. Unidad
Iztapalapa (UAM-I) in Mexico City.
The aim of this proposal is to teach students how to analyze with experiments, the problems
that they solve in their theoretical courses. The idea is to integrate Mechanics I, Waves and
Rotations and Differential Equations courses with topics that must learn in their experimental
courses, such as the use of spread sheets and video analysis. This proposal is not resolved in
textbooks and students enjoyed the experience.
Students observed and described the behavior of three different material balls falling freely
until they arrive to the floor. We consider that learning strategy would be very useful for our
experimental courses.
Keywords
University education, Laboratory activities, Teaching strategies, physical modeling, theoretical
analogy, simulation phenomena.
INTRODUCTION
The presented work was carried out in the course “Experimental Method II”, which is part of
science and engineering curriculum at the Universidad Autónoma Metropolitana – Iztapalapa
(UAM-I). Fernando Yañez Barona was one of the students that participate in this experience.
At this level, students know about mechanical waves and rotations, basic calculus and begin to
solve basic differential equations. In this course, students learn to analyze experimental data.
Term in the UAM-I lasts only three months so time students spend in the laboratory must be
use wisely to get the best learning. At the UAM-I, the physics experimental courses are
independent from theoretical courses, so students usually do not relate problems they solve in
theoretical courses with the work in the laboratory. Students feel that this is a very technical
course where they only measure and analyze experimental data, without understanding the
physical phenomena involved in the experiments. They feel disappointed and therefore they
underestimate curricular value of experimental courses.
In this work, we designed a research sequence on physics taking into account the objectives of
the course and giving students fulfillment and meaning to the experimental courses as well as
the theoretical ones. In this case, students were requested to use their theoretical knowledge to
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explain the how different types of balls bounce on the ground and on their desks, and at the end
they were capable of propose a mathematical model in order to explain the phenomenon.
The didactic goal was to involve students in an experiment that might seem simple but include
physical, mathematical, computer and multimedia tools. Students were free to reach the results
they could or they wanted.
EXPERIMENTAL METHODOLOGY
Students had to observe the movement of three balls (a sponge ball, a ping-pong ball, and a golf
ball) bouncing on the ground. Firstly, they listened to bounces until they stopped. Secondly,
they drew the path of each ball as they saw it moving. Consequently, they suggested hypothesis
for the consecutive bounces and they proposed the variables to be measured, how to measure
them, why they chose those variables and how they would use them in case they needed them
in the next analysis. They repeated the experiment but over their desk and finally they replaced
the balls by their notebooks which also fell freely from the same height. Obviously, they
noticed that the notebooks did not bounce.
After this qualitative analysis, with the help of the teacher, they decided to measure how high
the balls went up after each bounce and the time between bounces. In order to measure
distances, some students chose a tape measure, others chose a one-meter rule. For measure
time, they used chronometers. With these data they plotted the height versus time. In the first
instance, they believed that their measurements were the best they could get, but afterwards
they realize that only the initial height, which they fixed, was easy to measure, while for the
next heights were difficult to obtain with high accuracy, high precision and repeatability. They
were also surprised to notice that the heights reached depended on the type of ball and that in
order to get a graph similar to the one they drew before, they needed more data. For time
measurements, they found that as balls heights were smaller, time was more difficult to obtain.
At this stage, the principles of good measurements could be taught. There was a discussion of
measurement uncertainty followed by a lecture of the procedures for quantifying and reducing
measurement errors. Although, in the following stages, as the objectives were different,
students did not always calculate the measurement uncertainties.
The next stage was then to suggest other ways for measure the variables chosen in order to be
able at the end, to model collisions, of the balls and the notebook, with the ground and desk.
Then the teacher proposed to record the movement with their cell phones and to use the image
and video analysis package Tracker, then analyze how the path of the ball changed with time
and thus to calculate its speed.
The students acquire skills using their laptops, tablets and cell phones. From the graphs, they
learned that with Tracker, they could visualize the path of the movement of ball and obtain the
maximum heights as a function of time (Figure 1). Actually, the graph of maximun heights
versus time (h vs t) is a discrete one, were points are disconnected one from each other. But, the
students noticed that in that graph, if they interpolated the points, as it was a continuous
phenomena, and they connected all with a line, they had a nonlinear behavior similar to an
exponential decay that could be analyzed with Excel software in order to find the mathematical
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relation between maximum heights and time. At this stage, students reinforced their knowledge
about graph analysis.
Figure 1. Plot Height versus time for n successive bounces obtained with Tracker
By the other hand, from their theoretical courses, the students had already learned the law of
conservation of energy and they knew how to analyze the movement of an object that falls and
hit the ground. Here, they noticed that the object (in this case the balls), bounced several times
at hitting the ground and at each bounce, the maximum heights reached were smaller, so they
were aware that the maximum heights reached by the balls decreased because of the energy lost
during the bounces. Then with the h versus t plot, they could analyze the lost of energy due to
bounces.
In their theoretical courses, the students also had studied about elastic and inelastic collisions,
so they proposed to calculate the ball speed when it hit and when it left the floor. At this stage
of the research, they began to introduce the energy conservation to obtain the speed.
In order to be able to model the complete phenomena, as a hint, the teacher reminded the
students the theoretical problem of a body that slides without friction on an incline plane that
compress a spring at the end of the movement. How far the body goes back when it collides
with the spring at the base of the plane? All students considered that the body reached always
the same height. Changing the plane angle until 90 degrees, the body freely fall and bounce
when strikes the spring. The aim of the hint was to guide students to model the experiment they
were doing, simulating the ball bounce phenomena with mechanical elements such as springs
and bumpers. With this analogy and the h versus t plot in mind, and after reading about the
damped harmonic oscillator, they could already deduce that they could compare the ball
bounces to a mass-spring-damper system.
Then, the teacher proposed to determinate the force that pushed the ball up and to find why
their notebooks did not bounce. He also, asked them, knowing the initial height, to predict the
height the balls reached after the next bounces. From concepts learned on theoretical courses,
the students proposed to obtain the linear momentum before and after each bounce.
From the second Newton’s law, in the case mass-spring-damper, the net force can be related
with the rate of change of linear momentum and from the third Newton’s law, forces applied on
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the ground gives as a result a force equivalent to the normal force that push the ball up. By the
other hand, the normal force is distributed in the contact area if the ball undergoes a big
deformation, the contact time is longer and the normal force per area decreases and the force up
is small, but if the ball undergoes a small deformation, the contact time is shorter and the
normal force per area increases and the force up is very big. This observation explain why
notebook doesn’t bounce as a small ball.
When the students had already their own mathematical model, they could compare with other
models found in literature or in websites. For instance, in the website
www.sc.ehu.es/sbweb/fisica_/ of Angel Franco Garcia, the multiple bouncing of one ball is
described. In the website there is an analysis of the ball height with the number of bounces.
García found an iterated relation for each bounce and then he extrapolated to the case of a
temporary relation. At this stage, we review the concepts of kinetic and potential energies,
linear momentum, impulse and energy conservation law
Finally, the students realized that the essential difference between force and energy was so clear
in this case, since the weight of each ball did not change, however the potential energy,
dependent on height, varied according to the behavior of the coefficient of restitution and not of
falling force, as this is always the same.
THEORETICAL ANALYSIS In this section we briefly describe the process followed in the operational part of the
experiment. Here we explain the mathematics that we used; with the assumption of inelastic
collisions between the ball and the floor.
Restitution coefficient
It is known that in a frontal collision of two solid spheres, such as those billiard balls, balls
speeds before (v1, v2 ) and after (v′1 ,v′2), the collision are related by the expression (Hibbeler,
1997),
v′1 − v′2 = −ε(v1 − v2)
(1)
This relation was proposed by Newton, and ε the coefficient of restitution (figure 1). The
students used this theoretical result, but they did not, experimentally, confirm it, but if the term
was longer could be a nice experiment to be done at the classroom. Figure 2. Elastic shock in a plane.
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Figure 3. Inelastic shock of a ball hitting the ground while free falling.
In this case the floor do not move (v2 = v′2 =0).
Successive bounces a ball with the ground. At the ground, considering the ground as the second ball, v2 = v′2 =0. And the equation (1) is
reduced to (Figure 2),
v′1 = −ε v1
(2)
First bounce
Initially, the ball is at rest in a height h0, that is, gravitational potential energy is maximum. In
free fall just before the crash downward, the speed v1 must satisfy the energy conservation
principle:
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mgh = mv 2
2
(3)
According to equation (2), the ball speed after shock is v′1 = −ε v1, that is, the speed at which
the ball goes up and reaches a height h1, calculated by applying again the principle of
conservation of energy:
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1
mv '2 =mgh1 = mε 2 v12
2 1
2
h1 = ε 2 h 0
(4)
(5)
Second bounce
We can proceed in the same way for the second bounce, when the ball falls freely from the
height h1. The ball speed v2 before the second collision with the ground is calculated by the
principle of conservation of energy:
1
mv 22 =mgh1
2
(6)
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Again, equation (2) tells us that the speed of the ball after the second shock is v'2 = ε2v2. Then,
the ball goes up with the v'2 speed and reaches a maximum height h2 which is calculated using
the principle of conservation of energy:
1
1
mgh 2 = mv '22 = mε 2 v 22
2
2
(7)
Substituting equations (5) and (6) we arrive at
h 2 =ε 2 h1 = ε 4 h 0
(8)
Height versus time
Now it is easy to infer from the equations (5) and (8) that the maximum height, then of n
successive bounces, is given by
h n =ε 2n h 0
(9)
we must remember that 0≤ε≤1.
Required time for the ball to stop
The time the ball need to reach the ground when dropped from an initial height h0 from rest can
be calculated from the motion equation of a particle moving at constant acceleration:
1
y f = y 0 + v o t − gt 2
2
(10)
Where yf = 0, y0 = h0 and v0 = 0 i.e, we obtain that, once the ball hits the ground, it bounces,
goes to a height h1 and then falls back to the ground. The time it takes to go up and down is:
t1 = 2
2ε 2 h 0
2h1
=2
= 2t 0ε .
g
g
(11)
The ball bounces a second time and reaches a height h2, and falls back to the ground. The time
it takes to get up and down is:
t2 = 2
2ε 4 h 0
2h 2
2ε 2 h1
=
=2
= 2t 0ε 2
g
g
g
(12)
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Then it is clear that the time the ball takes up and down after the n bounce is
t n = 2t 0ε n
(13)
Here, we can use this expression to calculate the total time after infinite bounces as the sum of
the individual times between the bounces; i.e.
t∞ = t0 + t1 + t2 + · · ·= t0 [1 + 2ε (1 +ε+ · · ·)]
(14)
The terms between round brackets is a geometric series then,
#
2ε &
t ∞ = t 0 %1+
(
$ 1− ε '
(15)
That is,
t∞ = 2
h 0 # 1+ ε &
%
(
g $ 1− ε '
(16)
Note that if ε = 0 the time the ball needs to stop is exactly t0, which means that the ball loses its
energy in the first collision with the floor. But if ε = 1, the ball bounces indefinitely (t∞⇒ 1)
since there is no energy dissipation. The time tn, the ball remains in the air between n and n + 1
bounces at the ground is given by equation (13) which can be written as
ln tn = n ln ε + ln(2t0)
(17)
Where we have taken the natural logarithm of both sides of the equation (13). Equation (17) is
the equation of a line of the form y (x) = mx + b, (figure 4)
Figure 4. Logarithm of time versus numbers of bounces
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This means that if we plot ln tn versus n, we get a straight line of slope m = ln ε and y-intercept
b= ln(2t0), with t0 given by equation (13). If m is determined from experimental data (figure 4),
we can calculate the coefficient of restitution
ε = em
(18)
It is noteworthy that ε is important in determining how much energy is dissipated by the
bounces
Damped harmonic oscillator model
In the experiment, three different material balls (a sponge ball, a ping-pong ball, and a golf
ball), were dropped from the same initial height. Firstly, they were dropped on the ground and
after on the students desk. For each ball the experiment was repeated 10 times. On the ground,
the students observed that the behaviors of the ping-pong and a golf balls were very similar, in
spite of their different mass and textures. The sponge ball, which the smooth one, reached a
smaller heights and bounced less. On the desk, the behaviors of the ping-pong and a golf balls
were also very similar, as well as the sponge ball reached a smaller height and bounced less.
But, in this case, the sponge ball, reached higher heights and bounced more than when it hit on
the ground.
The students were astonished to found theses behaviors of the balls, and they wanted to explain
what happened. Their physical knowledge at this level of their studies, is not enough to
understand what really happened when they change the material, but if we use a simple
analogy using the theoretical concepts they know, it is possible to obtain a reasonable
explanation of the phenomena, as well is a nice to introduce students to the art of model
physical phenomena. Then, the teacher proposed them to consider that the balls behave like a
mass-spring-damper system (figure 5) and we present here the mathematical considerations.
Figure 5. Balls bouncing from the ground is modeled as a combined system mass, spring and damper
From the equation of motion of a damped harmonic oscillator (French, 2003), we have,
hn(t) = h0e−γt
(19)
where, γ is the damping coefficient. e-γt represents the envelope of the oscillation and it depends
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on the rate of decay of the amplitude, that is, it is associated with the loss of energy.
As we are considering that the ball behave like a spring-mass-damper system, if we compare
the last expression, with the one obtained from the curve height vs time plot where h can be
related with the coefficient of restitution (equation 9), and if hn is the maximun height after the
nth bounce (Figure 6), we obtain,
−γt = 2n ln ε
(20)
Actually, we must remain that the plot of maximun heights versus time (h vs t) is a discrete
plot, were points are disconnected one from each other and that we arrived to this analogy from
the students proposal of interpolate the points, as it was a continuous phenomena, and to outline
all with a smooth curve, in order to have a nonlinear behavior of the loss of energy during time,
similar to an exponential decay.
Then, the energy can be written as function of the number of bounces as follows:
E(n) = mgh 0e−2n log ε
(21)
Figure 6. Height versus bounce. Comparison of theoretical data (from model) and experimental data.
CONCLUSION
It is important for us to highlight that to use examples that contain a lot of physical concepts
can be reinforced by experimental practice. The participation of the students was markedly
enhanced with the feeling that each class was a challenge for them. The cooperation between
the students from different majors made each one to use their skills in order to obtain the best
results. But as the same time, it was really amazing, to experience the competitive atmosphere
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between work groups of students during one month and a half, 6 hours a week. Between,
students interested on study a physics major, were very interesting debates of the phenomenon,
while students interested on study an engineering major were more practical.
The use of software and computers as well as multimedia through their cell phones was an
important knowledge acquired to achieve the goal. Although, at the end, it was more important
the model than the measurement errors and uncertainties, at the beginning of the experience
there was a nice analyze of them.
The aim of the experience would not be reached if the students did not understand the physical
concepts so close as force and energy and how they are related with a non conservative
systems. They learned how to use simple analogies and they could build a model with which is
was possible to have a reasonable explanation of what they observed in the experiment,
although they have not yet the knowledge of the study of the deformation of solids.
They realized how changing the features of a mass-spring-damper system, could simulate the
different behavior when the material of ball was changed. They understood that if an elastic
material (sponge ball) is pressed against the floor, the contact area increases. Then pressure
decreases, the force that does not change, is not anymore applied to a point but to a bigger
contact area, giving as a result a less bounce. This is also the reason why the notebook does not
bounce.
The students also could associate the restitution coefficient with the different materials, and
they understood that how inelastic and elastic collisions depend on it.
References
French, P. A.. (2003) Vibrations and waves. MIT Introductory Physics series. CBS Publisher and
Distributor. USA
Hibbeler, R.C., (1997). Engineering Mechanics: Dynamics. 8th ed. Prentice Hall College Div. USA.
www.sc.ehu.es/sbweb/fisica_/ of Angel Franco García
Affiliation and address information
Armando Cuauhtémoc Pérez Guerrero Noyola and Fernando Yañez Barona
Departamento de Física.
Area de Mecánica Estadística
Universidad Autónoma Metropolitana -Iztapalapa
San Rafael Atlixco 186, Col. Vicentina,
09340 México, D.F.
[email protected]
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