CURVATURE

Week 9

In this week after I know to find the arc length, I learn how we find the curvature that means the measure of how sharply a curve bend.

 

 The circulars the same curvature at any point. Also, the curvature and the radius of the circle are inversely related. This is a circle with large radius has a small curvature, and a circle with small radius has a large curvature.

Many modern roller coasters features loops. Roller coaster loops are often not circular. If we looked closer at roller coaster loop, we will notice how the top is look like a half-circle, whereas the bottom looks different, with an increasing radius of curvature closer to the ground.

 

 

Arc Length of a Curve in Space

 

Arc Length of a Curve in Space

Week 8

In this week I learned how to calculate the arc-length of a space-curve. For example the arc length of helix.  A helix is also known as a cylindrical coil or solenoid coil. In mathematics, a helix is a curve in 3-dimensional space.

 

 To find the total length of the wire that is used to make the spiral helix of the spring, we need to know the axial height of the helix (h), the diameter of the helix (d), the number of turns in the helix (n).

The formula for arc-length of a helix or arc-length of a space-curve is  

 L = sqrt((n*Pi*d)^2 + h^2)

I noticed that the equation of calculating the arc length for a curve in space is very similar to calculating the arc length for a curve in the plane. We just need to add a z term to the formula for the arc length of a plane curve.

 

POSITION VECTOR FOR PROJECTILES

POSITION VECTOR FOR PROJECTILES

Week 7

In this week regarding to use calculus for designing a roller coaster? I found some information about “Derivation of the position functions for projectiles of mass m when it launched from an initial position r_0 with an initial velocity v_0 ".

 

 

We know that the acceleration a in the time t is:

a(t) = -gj

Where:

g is the earth gravity

Then if we integrate twice:

v(t) = ∫a(t)dt = ∫-gj dt = -gtj+ _{C_1}

r(t)= ∫v(t)dt = ∫(-gtj+_{C_1})dt=-1/2 gt^2 2j+_{C_1 t+C_2}

We know that: v(0) = _{v_0}, and r(0) = r_0, to solve for the constant vectors _{C_1}  and _{C_2}, doing this produces: _{C_1}  = _{v_0},and _{C_2}  = r_0. Therefore, the position vector is:

r(t)= -1/2 gt^2 j + t_{v_0}  + r_0      Position vector

Now I know how to find the ‘Position vector”. It helps me to find the position of the costar in any time when it works. 

 

 

 

 

 

 

 

ROLLER COASTER

ROLLER COASTER

Week 6

In this week I am looking for resources to do a research about how we can use calculus to design a roller coaster, for example; the “BATMAN: The ride” . I know from physics and math classes that the coaster's initial ascent is to get a potential energy. As the coaster gets higher in the air, gravity can pull it down a greater distance. This potential energy can be released as kinetic energy which is the energy of motion that takes the coaster down again by gravity force. Gravity applies a constant downward force on the coaster.

 

The roller coaster's energy is constantly changing between potential and kinetic energy. At the top of the curves of the first loop, there is maximum potential energy because the coaster is as high as it gets. As the coaster starts down of this loop, this potential energy is converted into kinetic energy. The coaster speeds up. At the bottom, there is maximum kinetic energy and little potential energy. The kinetic energy propels the coaster up again to the second loop, building up the potential-energy level. So the coaster tracks serve to channel this force. The math calculation control the way the coaster cars fall. If the tracks slope down, gravity pulls the front of the car toward the ground, so it accelerates. If the tracks tilt up, gravity applies a downward force on the back of the coaster, so it decelerates. Currently I am looking for resources to find mathematic calculation to design loops that will make everybody safe.