Maths and Physics

THIS EBOOK WAS PREPAREDAS A PART OF THE COMENIUS PROJECT

WHY MATHS?WHY MATHS?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 theinformation contained therein.

I.I. MATHS AND PHYSICSMATHS AND PHYSICSII.II. MATHS APPLIED TO PHYSICSMATHS APPLIED TO PHYSICS

III.III. KINEMATICS AND DYNAMICSKINEMATICS AND DYNAMICSIV.IV. HOW TO SCORE A GOALHOW TO SCORE A GOALV.V. PHYSICS OF SAILINGPHYSICS OF SAILING

VI.VI. REFRACTION OF LIGHTREFRACTION OF LIGHT

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:

F⃗ g=m g

F⃗ g=10 kg⋅5 m / s2=50 N

F⃗ r= F⃗a=ma⃗F r=5 N

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 PhysicsAs 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 DynamicsSubject: 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 ;DistanceWe 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.

AccelerationWe all know that the acceleration is calculated from the expression: a= v

t

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 ForceThe 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:

NewtonNewton''ss 1 1stst lawlawTheThe firstfirst lawlaw ofof motionmotion isis calledcalled thethe LawLaw ofof InertiaInertia. . ItIt statesstates thatthat “ “anyany objectobject atat restrest, , willwill tendtend toto staystay atat restrest, , andand anyany objectobject inin motionmotion, , willwill tendtend toto staystay inin motionmotion unlessunless actedacted onon byby anan unbalancedunbalanced forceforce.”.”

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.

NewtonNewton''ss 2 2ndnd lawlawNewton’s second law states that “The change in velocity (acceleration) with which an objectNewton’s second law states that “The change in velocity (acceleration) with which an object moves is directly proportional to the magnitude of the force applied to the object and inverselymoves is directly proportional to the magnitude of the force applied to the object and inversely proportional to the mass of the object.”proportional to the mass of the object.”

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.Newton's 3rd lawNewton's 3rd lawNewton’s final law of motion states that “for every action, there is an equal and oppositeNewton’s final law of motion states that “for every action, there is an equal and opposite reaction.”reaction.”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 ballWhen you kick a soccer ball, multiple things happen.

First of all, your leg is putting kinetic (or “movement”) energy into the ball. The formula for this is:

K E=12

m v2

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=v leg⋅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

IntroductionSailing gives many examples of Maths and Physics: Newton's laws, vector operations, Bernoulli's law, Archimedes' principle and many other applications.

The sailboatFirst of all, let's have a look to a sailboat and to its parts.

An introduction to vectors theory and operationsA 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=√ Ax2+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

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 b⃗ and c⃗ , 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 b⃗=n a⃗ : 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= a⃗⋅⃗b : the result is a number depending on the mutual position of a⃗ and b⃗ and

we define it as c= a⃗⋅⃗b=a bparall=a bcosθ where b cosθ is the component of b⃗ along the direction of a⃗ . This operation is commutative, as cosine is an even function, so a⃗⋅⃗b= b⃗⋅a⃗

• cross product c⃗=a⃗×b⃗ : the result is a vector whose magnitude is c = a b sinθ and whose direction is perpendicular to the plane where a⃗ and b⃗ 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

It is an anti-commutative operation, as sinθ is an odd function: so a⃗×b⃗=−b⃗× a⃗

Sailing and wind directionPhysics 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.

Right hand ruleLeft hand rule

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 =

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 p2 ΔV – p1 ΔV = 12

m v22−1

2m v1

2−m g h2+m g h1 Dividing by ΔV and rearranging the equation we

have: mv1

2

2 ΔV–

m g h1

ΔV− p1=

m v22

2 ΔV−

m g h2

ΔV−p2 thus we can derive Bernoulli's law as ρ=m

V

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 sailboatWhen 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

F⃗=m∑ a⃗ i and the acceleration is a⃗= Δ v⃗Δt

, 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-Componentshttp://www.physics.unsw.edu.au/~jw/sailing.htmlhttp://www.real-world-physics-problems.com/physics-of-sailing.htmlhttp://www.unc.edu/~thriveni/sailing/lift.htmlhttp://ffden-2.phys.uaf.edu/211_fall2002.web.dir/josh_palmer/basic.htmlhttp://scienceworld.wolfram.com/physics/BernoullisLaw.htmlhttp://adventure.howstuffworks.com/outdoor-activities/water-sports/sailboat4.htm

Refraction of lightRefraction 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 refractionLight 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=cv1

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 depthAn 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

MiragesMirages 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.