Maths and Physics

THIS EBOOK WAS PREPARED
AS A PART OF THE COMENIUS PROJECT
WWHHYY MMAATTHHSS??
by the students and the teachers from:
BERKENBOOM HUMANIORA BOVENBOUW, IN SINT-NIKLAAS ( BELGIUM)
EUREKA SECONDARY SCHOOL IN KELLS (IRELAND)
LICEO CLASSICO STATALE CRISTOFORO COLOMBO IN GENOA (ITALY)
GIMNAZJUM IM. ANNY WAZÓWNY IN GOLUB-DOBRZYŃ (POLAND)
ESCOLA SECUNDARIA COM 3.º CICLO D. MANUEL I IN BEJA (PORTUGAL)
IES ÁLVAREZ CUBERO IN PRIEGO DE CÓRDOBA (SPAIN)
This project has been funded with support from the European Commission.
This publication reflects the views only of the author, and the
Commission cannot be held responsible for any use which may be made of the
information contained therein.

II.. MMAATTHHSS AANNDD PPHHYYSSIICCSS
IIII.. MMAATTHHSS AAPPPPLLIIEEDD TTOO PPHHYYSSIICCSS
IIIIII.. KKIINNEEMMAATTIICCSS AANNDD DDYYNNAAMMIICCSS
IIVV.. HHOOWW TTOO SSCCOORREE AA GGOOAALL
VV.. PPHHYYSSIICCSS OOFF SSAAIILLIINNGG
VVII.. RREEFFRRAACCTTIIOONN OOFF LLIIGGHHTT

Maths and Physics
Mathematics is a very ancient science and it’s still the most widely studied. Maths is present in every
single science, without it, many knowledge that we currently have about the world would never be
found. In this work, we intend to illustrate and examine this idea. To do that, we will use physics, and
through 11th grade’s knowledge, show the importance of mathematics.
In Physics, Mathematics has a major importance because it allows us to calculate and equate
events of our everyday lives. From the fall of a rock, to the motion of a satellite, there are many
elements that can be translated into numbers (applied forces, mass, acceleration, velocity, mechanical
energy). This translation into paper, allows us to control and prevent unexpected events to happen.
In the image above, we can see an object that was released on an inclined ground without any
friction until it reaches soil. Nonetheless, from simple data, such as the value of the mass, the
inclination of the object, the distance to the ground and some knowledge of mathematical equations,
it’s possible for us to calculate many conclusions, such as the time that would take until the object
reach the ground, where would he eventually stop and the value of the forces that are being implicated
in the object.
Assuming that:
m= 5kg (mass)
distance of the object to the soil= 10m
inclination= 30º
There is friction force, on the value of 5N, after the object reaches the ground.
How to proceed to take advantage of the maximum amount of information:
⃗ Fg=mg

⃗ Fg=10 kg⋅5m/ s2=50 N
⃗ Fr= ⃗ Fa=m⃗a
Fr=5N
And through all this mathematical equations we can predict how the object will behave. So, the
object will slowly slide down the inclined ground with an acceleration of . Then it will reach the
ground with a velocity of , after of been dropped. Before the object stop completly, it will
still travel , for .
As it was previously shown, Math it’s an important factor in Physics. Nonetheless, this was only an
example. In Physics, Mathematic is important to calculate and to help to understand all kind of
things (boiling water, the fall of a stone, the movement of the planets, the light and sounds etc.).
Physics is just a example of many other sciences that need Math, because it is the base of many
others. It’s the logic behind the reason.
Maths applied to Physics
As we just said, Maths and Physics are strictly related subjects and often teachers of Maths teach
Physics too.
This presentation gives evidence to the link between these two subjects.

Kinematics and Dynamics
Subject:
In this work, we will talk about an experiment that is performed by all people, whether on foot, by car
or bicycle. The speed of a car is measured at each instant (in this case), values which are listed in the
graphic below. In this work we’ll explain what can we calculate from a graphic velocity/time and also
knowing that the motion was rectilinear uniformly accelerated.
The question is:
If I see a graphic of velocity/time, what can I calculate with the data provided?
This graphic represents the movement made by a car in 9 seconds in a uniformly varied rectilinear
motion. It’s very important to know that the movement is rectilinear because if it wasn’t, many
expressions would be different.
We all know the expression: v=dt
In the same way, d=v t ;
Distance
We have the two variables to complete the equation above in the graphic: the velocity and the time. If
we do the area of the graphic we find the distance because we are doing v t.
A.: The distance traveled by the car was 17,5 meters.

Acceleration
We all know that the acceleration is calculated from the expression: a=vt
If you notice, the slope of the graph is equivalent to this expression, because it’s also
vt
Using the 1st expression, the acceleration in the 1st section is m/s
Also, if we use the slope of the graphic in the 1st section, the acceleration is m/s. We can see that
is evidence that the slope of the graph also gives us the acceleration value.
In the 2nd section the acceleration is 0 because the slope is 0, in the 4th section the acceleration is
m/s, in the 5th section is 0 because the slope is also 0 and in the 6th section the acceleration is -
m/s. In this case, the acceleration is negative because it decreases the velocity value.
Net Force
The first law of Newton states that, if the net force (the vector sum of all forces acting on an object) is
zero, then the velocity of the object is zero or constant.
In the same way, if we see in a graphic velocity/time that the module of the velocity is changing, in
other words, there is acceleration, must exist a net force.
In the 1st, 4th and 6th sections, the net force is nonzero because, like I said, the velocity module is
changing.
In the 2nd and 5th section, the net force is zero because the velocity is constant.
This is also proved in the second law of Newton. Newton's second law says that the net force is equal
to the product of the mass and acceleration. We can conclude that if everything has a mass, then if the
body has acceleration, it will have a net force.
Through this study, we were able to realize that, from a speed/time graph, we can know the
distance, the acceleration and the net force. Obviously, we can also know the velocity in a certain
moment looking directly to the graphic.

How to score a goal
The video shows a free kick shot by Roberto Carlos in 1997; the ball follows a tremendous curve and
with the high speeds gets into the post with the astonishment of all audience and above all the of the
poor goalkeeper. How did it happen? Was it a sort of magic? Of course not! There is a specific
explanation according to the basic rule Physic. Let’s see how it works.
First of all we remember the Newton’s laws, the bases of any physical reasons, opportunely applied to
the football’s Physic:
NNeewwttoonn''ss 11sstt llaaww
TThhee ffiirrsstt llaaww ooff mmoottiioonn iiss ccaalllleedd tthhee LLaaww ooff IInneerrttiiaa.. IItt ssttaatteess tthhaatt ““aannyy oobbjjeecctt aatt rreesstt,, wwiillll tteenndd ttoo ssttaayy aatt
rreesstt,, aanndd aannyy oobbjjeecctt iinn mmoottiioonn,, wwiillll tteenndd ttoo ssttaayy iinn mmoottiioonn uunnlleessss aacctteedd oonn bbyy aann uunnbbaallaanncceedd ffoorrccee..””
This unbalanced force could be: gravity, wind, or any moving object. In the Physics of soccer
unbalanced force is usually the player’s foot. The player will use muscle in the body to create a force to
move the leg and kick the ball.
Because the ball is at rest, it will continue to stay at rest but once kicked, it will keep moving in a
straight line without any intent of stopping because of the Physics of soccer.
NNeewwttoonn''ss 22nndd llaaww
NNeewwttoonn’’ss sseeccoonndd llaaww ssttaatteess tthhaatt ““TThhee cchhaannggee iinn vveelloocciittyy ((aacccceelleerraattiioonn)) wwiitthh wwhhiicchh aann oobbjjeecctt
mmoovveess iiss ddiirreeccttllyy pprrooppoorrttiioonnaall ttoo tthhee mmaaggnniittuuddee ooff tthhee ffoorrccee aapppplliieedd ttoo tthhee oobbjjeecctt aanndd iinnvveerrsseellyy
pprrooppoorrttiioonnaall ttoo tthhee mmaassss ooff tthhee oobbjjeecctt..””

Using the Physics of soccer this simply means that if the ball has a lot of mass, it will require more
force to accelerate and if the ball has little mass, it will require very little force to accelerate when
the soccer ball is kicked.
NNeewwttoonn''ss 33rrdd llaaww
NNeewwttoonn’’ss ffiinnaall llaaww ooff mmoottiioonn ssttaatteess tthhaatt ““ffoorr eevveerryy aaccttiioonn,, tthheerree iiss aann eeqquuaall aanndd ooppppoossiittee
rreeaaccttiioonn..””
This literally means that if you kick the soccer ball, it will kick back at you just as hard.
The only reason you don't feel or realize this, is because our legs have more mass, meaning more
inertia, which is the resistance to move according to the Physics of soccer.
Kick a ball
When you kick a soccer ball, multiple things happen.
12
First of all, your leg is putting kinetic (or “movement”) energy into the ball. The formula for this is:
KE=mv2
Here the kinetic energy equals one half of the mass of your leg multiplied by the velocity of your leg,
as it hits the ball, squared. Another thing that happens when you kick the ball is that the ball deforms.
The side of the ball that your foot strikes becomes flat for just over 0.01 seconds. The energy going in
to the collision is the kinetic energy of your foot plus the stored energy in the deformed ball; the energy
coming out is the kinetic energy of the ball plus some heat.
The more the ball deforms the more energy is lost to heat.
The conservation of energy law causes the ball to go faster than the velocity of your foot! The actual
formula for the velocity of the ball is:
vball=
vleg⋅mleg
mleg+mball
⋅(1+e)
The “e” in this formula is called the coefficient of restitution and measures what speed a ball bounces
up at compared to the speed it hits the ground on the way down. This ranges from 0 to 1, with zero
being that the ball does not bounce up at all, and one being that the ball bounces to the
same height every time. This can also be measured by taking the square root of the ratio of the height
of the bounce to the original height.

Usually a official ball dropped from 2 m bounced up to a height of 1 m, so ≌0,7
Relevant data :
Before considering the applicable formula, let’s individuate the parameters involved.
The main cause of swerve is the air’s friction.
Others elements of consideration are:
The speed (V); the radius of the ball (R) and its mass (m); the air pressure (ϱ); the angular speed (ω);
the linear distance run before swerving (χ).
We admit that for purposes we are not considering the wind and the variations of the air pressure during
the run.
The given data are the followings :
- V= 30 m/s;
- R= 0,11 m;
- m= 0,42 kg;
- ϱ= 1,3 kg/m3;
- χ= 30 m;
The applicable formula of Physics are basically two:
1) The so called “Magnus effect”
2) 2) the Bernoulli’s equation.
Let's have a look to these Physics laws:
1) When a solid begin to rotate in a fluid (in our case the air) create a difference in the velocity of
the fluid at the surface of the solid between the upper and lower sides, due to the combination of
the angular velocity and the fluid flow velocity. This difference of velocities causes a pressure
difference, thus resulting in a net force.
2) In fluid dynamics, Bernoulli's principle states that for an inviscid flow, an increase in the speed
of the fluid occurs simultaneously with a decrease in pressure or a decrease in

the fluid's potential energy. in a steady flow, the sum of all forms of mechanical energy in a
fluid along a streamline is the same at all points on that streamline. This requires that the sum of
kinetic energy and potential energy remain constant. Thus an increase in the speed of the fluid
occurs proportionately with an increase in both its dynamic pressure and kinetic energy, and a
decrease in its static pressure and potential energy.
From the above principles we obtain
Where D is for the swerve.

Physics of sailing
Introduction
Sailing gives many examples of Maths and Physics: Newton's laws, vector operations, Bernoulli's law, Archimedes'
principle and many other applications.
The sailboat
First of all, let's have a look to a sailboat and to its parts.
An introduction to vectors theory and operations
A vector is an oriented segment that has a magnitude and a direction and it is represented by an arrow whose
direction is the same as the direction of the vector and whose length is proportional to the magnitude of the
vector.
Vectors have the following two properties:
• equality: two or more vectors are equal if their magnitudes and directions are the same
• the negative of a vector is a vector having the same magnitude but opposite direction.
Any vector can be expressed as the sum of two other vectors representing the
projection of the vector along two perpendicular directions, called vector's
components: the (scalar) component of a vector along a line is the length of
the projection of the vector on that line. The components of a vector depict the
influence of that vector in a given direction. The combined influence of the two components is
equivalent to the influence of the single two-dimensional vector: Ax = A cosθ; Ay = A sinθ
A=√Ax
2+Ay
2
A variety of mathematical operations can be performed with and upon vectors:
• addition : two vectors can be added together to determine the result (or resultant). We can use the
⃗c
Carnot's theorem a2=b2+c2−2 b c cosα (whose Pythagorean theorem is a particular case if the
vectors are perpendicular) where α is the angle
between ⃗b and , a, b and c are vectors'
magnitudes; to add two vectors we can use the head-tail
method or the parallelogram method

• subtraction : two vectors can be subtracted adding one to the opposite of the other. Using the
parallelogram method, we can see that the difference vector is the other diagonal than the one we use
to sum them
• product : there are 3 kinds of products involving vectors:
• product of a vector by a scalar number ⃗b=n⃗a : b magnitude n times the magnitude of a; if a is
positive, b has the same direction of a, else its direction is the opposite
• dot product c=⃗a⋅⃗b : the result is a number depending on the mutual position of ⃗a
and ⃗b and
we define it as c=⃗a⋅⃗b=a bparall=a b cosθ where b cosθ is the component of ⃗b along the
direction of ⃗a
. This operation is commutative, as cosine is an even function, so ⃗a⋅⃗b=⃗b⋅⃗a
• cross product ⃗c=⃗a×⃗b : the result is a vector whose magnitude is c = a b sinθ and whose
direction is perpendicular to the plane where ⃗a
and ⃗b are laying. We can use some practical
rule to know if the vector is coming out or getting into the plane: the right hand rule
Right hand rule
Left hand rule
It is an anti-commutative operation, as sinθ is an odd function: so ⃗a×⃗b=−⃗b×⃗a
Sailing and wind direction
Physics behind sailing is very interesting as sailboats don't need the wind to push from behind
in order to move. A sailboat can move as a result of the interaction between its sails and the
wind.
Its quite obvious understanding that if the wind blows straight forward, the sailboat goes
straight forward sailing in the same direction of the wind: this is done as the sails push the
boat forward. The wind is faster than the boat so the air is decelerated by the sails and the
sails push backwards against the wind: in this way the wind pushes forward on the sails,
causing the boat to move forward. But for a boat with normal sails, the issue is that,
downwind, you can only sail more slowly than the wind, even with a spinnaker. This is fine, but it
is a very limited use of a sailboat.

It might be difficult for one person realizing that the wind can be blowing from the side and the sailboat can still
move forward. Even more: while a boat sailing with the wind can never sail as fast as the wind, a boat sailing with
the wind at an angle can match or even surpass the speed of the wind.
How is this possible? The key to this is a phenomenon known as “lift”1 that
results from differences in pressure in the well-known principle of aerodynamic
lift: if you are a passenger traveling in a car moving along and if you place your
right hand out the window, you can see that if you tilt your hand clockwise your
hand will be pushed backwards and up. This happens as the force of the air has a
sideways component and upwards component, so your hand is pushed
backwards and up. The "lift" produced by a sail is mainly directed horizontally:
this word is used as the mechanism is the same as the one that produces lift on
an aircraft wing since both wings and sails are airfoils; the only difference is that
wings are usually oriented horizontally, while sails are normally oriented vertically.
So, the sail modifies the airflow around it vertically, the corresponding "lift" force is
oriented horizontally and the only difference between the wing and the sail is that
the base side of the sail is not filled in and there is a stationary pocket of air staying inside of the curve of sail, so
that as the wind blows by it, it passes by the sail just like in the diagram below (the grey area is a stationary pocket
of air).
Sailing against the wind
A sailboat can move in the opposite direction of the wind as:
• the sails can change the direction of the wind to create a thrust
• Bernoulli's Principle2 (also called the Longer Path Explanation)
If the boat is facing the wind, for instance, approximately with 45° angle, the
sail is kept straight with the boat and the wind flows into and over the sail: its
direction is changed as it follows the shape of the sail.
In this case we have three resulting forces:
• the drag caused by the wind moving over the sails
• the lateral force exerted on the sail while the sail changes the
direction of the wind
• the final direction and velocity of the air after being redirected.
1 To understand better the lift, have a look to this video: https://www.youtube.com/watch?v=aFO4PBolwFg
2Bernoulli's equation:
12
ρ v2+ρ g h+p=constant where: ρ = fluid density, v= fluid flow velocity, g =
12
12
acceleration of gravity, h = height above a reference surface, p = pressure.
To understand this law, let's consider a pipe through which an ideal fluid is flowing at
a steady rate. Let's name W the work done by applying a pressure P over an
area A, producing an offset of Δl , or volume change of ΔV. Let's name 1 the first
point of the pipe section and 2 a further point.
The work done by pressure force is ΔW = ΔW2 – ΔW1 = p2 ΔV – p1 ΔV.
But for the kinetic energy conservation law, we have: ΔW = ΔK – ΔU
So pΔV – pΔV=mv2mv2 1 2
−1
2−m g h2+m g h1 Dividing by ΔV and rearranging the equation we
have: mv1
2
2 ΔV
–
mg h1
ΔV − p1=
mv2
2
2 ΔV −
mg h2
ΔV −p2 thus we can derive Bernoulli's law as ρ=mV

The drag (backward pull) between the sail and air is low and it is usually not considered in the schema. The
relevant forces are caused by the final boat velocity and the change between the initial and final wind velocities3:
this is however influenced by the boat's keel resistance into the water as the keel works against the water that is
moving slower to resist to any lateral forces. Thus, when you sail, the boat can move against the force of the wind
and it can move even faster than the wind velocity.
The particles on the outer side are traveling farther (as they follow a
curved trajectory) in the same amount of time, so they must have a
higher velocity than the particles on the other side: these higher-velocity
particles have more room to spread out, forming a low-pressure
area. On the inside of the sail, the slower air particles are
packed together more densely, creating a higher-pressure area. This
difference in the pressure on the sails acts as a forward suction,
producing lift: the air applies a force on the sail as it moves from a
higher pressure to a lower pressure.
So, when the wind flows over one side it fills the sail while the air
flowing on the other side is moving faster and cannot push as hard
and thus the sail receives a force that is normal to the direction of the wind: this normally would not push the
sailboat against the wind but, as we said before, the keel of the boat resists much of the lateral movement so that
the boat can move forward. The combined forces that are pushing the boat perpendicular to
the wind are greater than the force of the wind pushing the entire boat and sails backwards.
If the boat is moving with a constant velocity, this means that there is no acceleration, so the
force Fsails acting on the sails because of the wind must have the same magnitude and
opposite direction than the resultant due to the keel resistance FK and to the drag FD, as in the
schema.
It's impossible for a sailboat to travel directly into the wind as the resultant force Fsails has no forward
component. Instead, it has a backward component meaning the sailboat would travel backwards.
So there is an upper limit on how large θ can be. For very efficient sailboats this upper limit is around 60° .
The velocity of the wind relative to the boat (Vw) depends on the speed of the boat (Vboat) and we can calculate
it using vector addition with the formula Vw = Vw1 — Vboat , if we know Vboat and the wind velocity relative to the
water (Vw1).
The optimal wind angle for greatest sailboat speed is when Vw is blowing from the side, because the lift force is
pointing in the forward direction (parallel to the boat center line) and because the forward push force (forward
component of Fsails) remains fairly constant as Vboat increases.
But if the wind is blowing from behind the boat, Vw (and therefore wind force has the direction from behind the
boat) depends on Vboat: the faster the boat moves forward, the lower the relative wind velocity Vw and the lower
the wind force.
However, if the wind is blowing from the side it is actually possible for Vboat to be greater (in magnitude) than Vw
because the push force is great and constant enough to propel the sailboat to a high speed.
Titling of the sailboat
When the wind is blowing from the side the sailboat can tilt because of the torque created by the forces Fsails , FK ,
and FD: the sideways component of these forces, acting perpendicular to the center line of the boat, creates a
rotation of the sailboat. This torque is balanced by the counter-clockwise torque generated by the weight of the
sailboat and the buoyancy force of the water.
3 The forces increase as the velocity of the wind increases: for the Newton's first law we know that the force is
⃗F=mΣ⃗ai and the acceleration is ⃗a
= Δ⃗v
Δt , so if the velocity changes, there is an acceleration and thus there is
a force. However, the forward speed can become greater than the speed of the wind and this causes the boat to move: the
drag force will increase as the velocity of the boat increases and when it becomes equal to that of the forward movement it
only means that the boat cannot accelerate any further and that is the top speed.

If we name Fsails,h the sideways horizontal component of the force Fsails and FK,h
and FD,h the sideways horizontal components of the forces FK and FD we can
see that these component forces rotate the sailboat until the moment arm R
becomes large enough so that the weight and buoyancy forces are able to
stop the rotation: the result is a strong leaning of the sailboat.
http://www.physicsclassroom.com/class/vectors/Lesson-1/Vector-Components
http://www.physics.unsw.edu.au/~jw/sailing.html
http://www.real-world-physics-problems.com/physics-of-sailing.html
http://www.unc.edu/~thriveni/sailing/lift.html
http://ffden-2.phys.uaf.edu/211_fall2002.web.dir/josh_palmer/basic.html
http://scienceworld.wolfram.com/physics/BernoullisLaw.html
http://adventure.howstuffworks.com/outdoor-activities/water-sports/
sailboat4.htm

Refraction of light
Refraction is the bending of a wave when it enters a medium with different density, so its
speed is different. The refraction of light when it passes from a fast medium to a slow medium
bends the light ray toward the normal to the boundary between the two media. The amount of
bending depends on the indices of refraction of the two media and is described quantitatively
by Snell's law: when light passes between two media with different indices of refraction the
incident ray, the refracted ray and the normal lie in the same plane, and the two corners follow
this relationship: n1sin θ1=n2 sin θ2 or, in other words:
n1
n2
=
sin θ2
sin θ1
Index of refraction
Light can travel in vacuum but also in other
materials.
The index of refraction of a substance is a
measure of the speed of light in that substance.
Each material has a different index of refraction
which is the ratio between the speed of light in
vacuum (c), and the speed of light in the material
(v): n1= c
v1
The velocity at which light travels in vacuum is a
physical constant called c, and the fastest speed
at which energy or information can be
transferred.
In the table you can see the indices of refraction
of some different media:
An example of refraction in a glass full of water:

There are two main phenomena connected with the refraction:
• the apparent depth
• the total reflection
Apparent depth
An underwater object appears nearer the surface than it really is.
The apparent depth (d') is linked to the real depth (d) by the relation : d '=d (n2 /n1)
n1 is the refractive index of water (where the object is located)
n2 is the refractive index of air (where the observer is located)
Total Reflection

When the angle of incidence is greater than the limit angle the refracted ray is missing and all
the light is reflected.
The limit angle is calculated by this formula : sin x=
n2
n1
(n2 < n1) - x = limit angle
Mirages
Mirages are examples of total internal reflection. The conditions most likely to produce a
mirage happen when a layer of hot air lying immediately above the ground with cooler air
above it. This is quite usual during a Summer day as the ground becomes very hot.
Light rays from a distant object travel in a straight line through the cool air to the observer's
eye. But other light rays from the object travel toward the ground and come in contact with the
surface of separation between the cool and the hot air with their different optical densities: the
rays which struck this surface very obliquely (at an angle greater than the limit angle) would
be reflected upward again and thus reach the
observer's eye.
In this way the observer sees the distant object not only
upright but also inverted as though mirrored in a pool of
water.
This can happen in the desert but also on heated roads
during the Summer:the reflection of the sky and clouds
appears just above the surface of the road as though
mirrored in a pool of water.

Maths and Physics

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