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These slides were used for a talk to primary school students during the masterclasses at the University of Bath at the beginning of 2012. I really liked this one! :P
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What are we doing here?
We want to learn some maths!
So, what do the Angry Birds fit in all this?
Well, it so happens that we can find some maths in there…
What are these Angry Birds?
We haven’t started playing… when we already have some maths questions! What should I do to actually hit the pig? Should I pull just a little bit or maybe quite hard? Should I aim to the pig directly or should I aim high to hit him when I’m falling down?
We are now going to study some of the concepts needed to answer these questions. We are going to study the maths related to the branch of physics called mechanics that deals with the parabolic throw.
So, let’s get started! We have to learn 4 things this morning!
1. Cartesian plane and two dimensional vectors.
2. Horizontal movement with constant velocity.
3. Vertical movement with constant acceleration.
4. Mixing everything together! We want to see how a parabola looks like!
How far is my friend Carla from the Big Ben’s clock?
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Origin
Point in the plane
The Cartesian plane is a system of reference to specify each point in the plane uniquely by a pair of numerical coordinates.
The origin is our point of reference. We may choose it, but it most be fixed.
All other points in the plane are referred to the origin.
The line segment that goes from the origin to a given point in the plane is called a vector.
Every vector has a horizontal component (x-coordinate) and a vertical one (y-coordinate). These components are called rectangular components.
How do we create a Cartesian plane?
1. Set your point of origin.
2. Draw a line to the right, marking every unit length to create your “x-axis” or horizontal component.
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3. Now do the same but vertically to create the “y-axis” or vertical component.
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4. Draw the grid.
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5. To name a point in the plane, write between parenthesis the x and y coordinates (x,y). eg. (8,7)
(8,7)
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We may add vectors component wise, but multiplication is not defined. Negative components are also allowed which would correspond to a place below or to the left of the origin. (2,-3) + (4,1) = (6,-2)
Mathematicians work with vectors all the time! They even some times work with vectors in space with three components: length, width, and height.
Albert Einstein, said that space and time were no different one from each other, hence he used 4 dimensional vectors: (x,y,z,t). Even worse, string theorists, like Stephen Hawking, are now saying that our universe may consist of nearly 11 dimensions…
What is the average velocity of Usain Bolt if he ran 100 metres in 9.69 seconds?
displacement
time
Finding the velocity of an object moving following a straight line is quite simple: simply divide the total displacement, the total distance the object travelled, by the time it took to travel that distance.
What if we know the average velocity and the time, can we find the distance an object moved? Can we solve this formula for d?
What about the time variable?
Now, try the following experiment: Put an apple on top of a desk. See if it moves on its own horizontally…
Why do you think it won’t happen? Can we do something to make it happen? Can we make it move? How?
Now push it! It moves, right? See if it comes back…
Why do you think it won’t come back? What should happen for it to come back after you push it?
Sir Isaac Newton said a long time ago:
“Corpus omne perseverare in statu suo quiescendi vel movendi uniformiter in directum, nisi quatenus a viribus impressis cogitur statum illum mutare.”
This is Newton’s First Law of motion. In English this means that every object will keep resting or moving in straight line with constant velocity until there is a force that makes it change its current state.
Now throw the apple up in the air… why does it come back?
I wonder… why do they fall?
This is “la Torre pendente di Pisa” or Tower of Pisa for short.
Imagine a big apple at the top of it.
Let it go down, without pushing it or throwing it. Now, on mid air, is it going faster or slower than at the beginning?
What’s it’s velocity at the moment it’s not moving?
The change in velocity per unit of time is what we call acceleration. The force of gravity makes things have an accelerated movement. The acceleration of gravity on Earth is of
… going downwards.
Since acceleration makes moving objects change their velocity, formulas are a bit more complicated than the previous case where velocity was constant. So we’ll define the following concepts:
time
With a little help from Galileo, let’s analyse the experiment…
Initial position.
Initial velocity
Throw the apple and start timing.
time
Stop time!!
Position at time
Velocity at time Recall the gravity acceleration: Now, behold the formulas
for uniformly accelerated movement:
… and let the time continue.
Let’s go back to our previous example… We’re letting an apple fall down from the top of the Tower of Pisa, which we know it’s 59 metres above the ground. First question! After 3.46 seconds falling, where’s the apple?
Well, the initial position was 59 metres and the initial velocity was 0 metres per second, since it wasn’t moving. Following the formulas:
It was indeed about to hit the ground! Second question: what is the velocity of the apple at that moment?
A negative velocity means that the object is moving down. A positive one, that it’s moving up.
Now we’re ready for the real thing! Try to keep up because we’re using everything we’ve seen in here… ok? Here we go…
Where’s the bird?
Put a Cartesian plane with the origin on the bird, on the x axis the horizontal distance and on the y axis the vertical position..
Push the bird horizontally so that it has a constant horizontal velocity.
Horizontal velocity
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ty At the same time, push the bird vertically
so that it has a vertical velocity.
Start timing… and every few seconds find the horizontal distance travelled by the bird and its vertical position. Mark those points in the plane. You can do this on a table.
Join all the points to see the trajectory described by the bird: a parabola!
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Velocity is a vector and as such it has a horizontal component and a vertical one.
Horizontal velocity
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However, when we actually measure how fast we’re moving, for example inside a car with a speedometer, we’re measuring just one number: the speed.
Speed and velocity are related by the Pythagorean theorem:
We can actually find the velocity of an object given the speed, but we need another measurement: the angle of elevation.
However, this and other topics related to parabolic movement, mechanics or physics in general require more maths than what we can cover today…
