Deriving
Calculus Behind the 3 laws
Alexander Kareev
3rd hour
APPC
05/14/06
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( A picture of me
with guitar, Alex Kareev. Student of
This web page was created as part
of the curriculum for the AP Physics C (APPC) class. I am a student finishing
this class and I decided to cover this topic of Calculus and the 3 laws of
Calculus is very important for the AP Physics C course. Derivation and Integration of equations and graphs is a very important part of this course but there is also integration of formulas which is also important for physics. Being able to integrate and derive formulas and understand the relation between formulas will help you understand the relations between physical concepts such as force and energy for example .
This is a very big topic and I will not go to far in it and neither will I try to explain the integration and derivation, I will just use it in context of Newton’s 3 laws to show how it work and also explain their importance.
If you are conf used about calculus applied to formulas I will just give you a very popular derivation of formula
Her for example is a formula for Kinetic energy
KE= ½ mv^2
What happens if we teak a derivative of it (assume that m is a constant)
= ½ *2 mv = m *v
Which is the momentum
P= mv
So the instantaneous slope of the kinetic energy graph will give as momentum. And area underneath the momentum graph will tell as the displacement in kinetic energy.
You see how this is connecting the physics concepts using math.
Lets for example say there is a 1 kg ball and it is moving with changing velocity here is the data
v (m/s) |
KE (1/2
mv^2) (J) |
0 |
0 |
1 |
0.5 |
2 |
2 |
3 |
4.5 |
4 |
8 |
5 |
12.5 |
6 |
18 |
7 |
24.5 |
8 |
32 |
9 |
40.5 |
10 |
50 |
The instantaneous slope on each point would provide the momentum of the object
In the moment you will see how the 3 laws on
A brief history of
And here are some fundamental information about the calculus
and
You probably know the lame definition of the laws but here
is how
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Inertia is based on a simple
concept: an object in motion tends to remain in motion. An object at rest tends
to remain at rest. This is counter-intuitive to us, since we know that if we
slide an object along the floor, it will slow down. Similarly, something may
decide to fall off a shelf in a way that looks like it decided "all by
itself". But
The concept of inertia refers to
the tendency of an object to remain in motion, or to remain at rest. By
intuition we can see that the tendency is somehow related to mass. It is easy
to catch and stop a baseball; I don't recommend attempting to catch and stop a
freight train. It's also a lot harder to move a loaded filing cabinet than it
is to slide a single book. In order to measure how hard it was to overcome an
object's inertia,
[edit]
Force can be a confusing term
because it is an everyday term, as well as a physics term. When we use it in
everyday language, we use it in a whole number of different contexts, for
instance "He forced the door", "He was forced to take third
semester Calculus", or "This justifies the use of force". There
are a great many ways in which the word force may be used in casual conversation.
However, in physics, there is
only one meaning to force, that which was given to us by
where m represents the mass of the object in question and a its acceleration.
When you consider
What is important about this law
is that force is now quantified, and can be shared between two objects. For
instance I can design a weight that applies a force of thirty
[edit]
Now that we have the concept of
force, we can begin exploring how it effects the world around us. Consider
this. You are most probably sitting in a chair. How many forces are acting on
you at this moment? Since you are not accelerating in any direction it would be
easy to think that there are no forces, but that would be incorrect. Instead
there are a great many forces that are all balanced. For instance, there is the
force of gravity pulling you down. If you were magically suspended there above
the Earth, the force of gravity would pull you swiftly, and ungraciously, to
the ground. But you do not fall to the ground, instead you exert a force on the
seat of the chair.
Here is where
This is the force that you apply to the seat of your chair. However, thanks to
This is a hard concept, because
it always applies, even when you might think it does not. Let us deal with a
human attempting to catch a freight train. When the freight train hits the
human, which has more force applied to it? The answer is, from the title of
this section, that the forces are equal. The force applied to the human by the
train is the same as the force applied to the train by the human. Now, you
should note that a few thousand
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( the above information was copied from the http://en.wikibooks.org/wiki/Physics_with_Calculus:_Part_I:_Newton's_Laws
Website. I would like to rephrase it, but it is preaty compact and to the point. The site again is http://en.wikibooks.org/wiki/Physics_with_Calculus:_Part_I:_Newton's_Laws
. Go their to get the full picture)
Useful links :
http://home.case.edu/~sjr16/pre20th_europe_newton.html
http://en.wikipedia.org/wiki/Isaac_Newton
http://scienceworld.wolfram.com/biography/Newton.html
http://euler.ciens.ucv.ve/English/mathematics/newton.html
http://library.thinkquest.org/11902/physics/newton.html
http://www.niehs.nih.gov/kids/rdpartytx.htm
and the best page of all on this topic is:
http://en.wikibooks.org/wiki/Physics_with_Calculus:_Part_I:_Newton's_Laws