Eject based on mathematical concepts and ideas to explain how a roller coaster works and how does Differential Equations apply to it.

1. Roller Coaster
Differential
Equations
applying on Roller
Coaster
Lucas Machado de Vasconcelos
Differential Equations 7:00AM to 8:15AM

2. Principles of a Roller
Coaster
Roller coasters are driven almost entirely by basic inertial,
gravitational and centripetal forces, all manipulated in the service of a
great ride. Amusement parks keep upping the ante, building faster
and more complex roller coasters, but the fundamental principles at
work remain the same. Even though for the beginning of the ride it
has to have some type of propulsion or it has to start from the top of
something using its own gravitational force with the inertia of it going
on for the all ride.

3. What is the roller coaster compose
of?
The Roller Coaster is composed for many items, and most and all of
then are created to give support and protection to the ride, and they
are mainly: The track; The support for the track; The car;
The track-> On most of the places are made of steel because it gives
a better stability, stoping it from moving when it is very windy. Although
we still can find wood ones but those are usually smaller than the steel
ones because they are not as firm as thrones made out of steel.
The support for the track-> Are basically the aims that connect the
track on the ground giving it support in order to help it to stay at the
position that was expected without any problem.
The car-> They are usually the object that will transports passengers
around a roller coaster's circuit. On Roller coaster those cars are
connected by specialized joints which increases the car safety
principally on the loops.

4. Parts of a Roller Coaster:
-> Track.
-> Track Support
->Car or Train

5. What makes the Roller Coaster
move? Part-1
The Roller Coaster movement are based on its
own forces during the entire circuit, although as
every single vehicle it does needs a propulsion or
something to start the ride. On the case of a
Roller Coaster they use, an electric winch winds
the cars to the top of the first hill. Once at the top
of the hill this winch pulls the Roller Coaster
down, and right during this process the energy
for the entire circuit will be stored on the roller
coaster using it to move during he tracks even
when they have loops. All this energy stored
during the ride is called Potential Energy.

6. What makes the Roller
Coaster move? Part-2
This energy stored during some points of the circuit
will be transformed into Kinect Energy. According to
Newton’s First Law of Motion, “an object in motion
tends to stay in motion, unless another force acts
against it”, wind resistance or the Friction along the
track are forces that work to slow down the train.
So in theory with the potential energy acquired by
the roller coaster the ride could go forever and ever
although this energy on practice is constantly losing
its own magnitude for the Air resistance and
Friction.

7. Why don’t I fall out when a roller
coaster goes upside down on a loop?
Part-1
It’s all a matter of physics: energy, inertia, and
gravity.When you go around a loop, you feel pushed
against the outside of the car and that can be
explained by the force called Centripetal. This force is
the one that keeps you in your seat.
In the loop upside down, it’s Inertia that keeps you in
your seat. Inertia is the force that presses your body
to the outside of the loop as the train spins around.

8. Although gravity is pulling you toward the earth, at the
very top the acceleration force is stronger than gravity
and is pulling upwards, thus counteracting gravity. The
loop however must be elliptical, rather than a perfect
circle, otherwise the centripetal (g) force would be too
strong for safety and comfort.
How do we know whether a roller coaster is safe?
Engineers and designers follow industry standards and
guidelines. The first “riders” are sandbags or dummies.
Then engineers and park workers get to try it out.
Why don’t I fall out when a roller coaster g
Part-2

9. Illustrating the movement of a Roller Coaster.

10. How much height does it have to
be in order to complete the loop?
So during the circuit generally before a loop the
height of the track has to be at least 5/2 (2.5) of the
size of the loop in order for it to have force enough to
go around facing no problems during the loop and
having more than enough force in order to not get
stuck on the top of it or not even pass the first half of
the loop. (Calculation 1).