Contour Map

 

Contour Map

Week 12

Contour map is another name for a topographic map. It is a map that shows the elevation of land on a flat paper surface. The contour map contain an imaginary lines which connects points of equal elevation called contour line. Such lines are drawn on the plan of an area after establishing reduced levels of several points in the area.

 

For example, to sketch a contour line given equation  

Using level curves corresponding to:

k = 0, 1, 2, 3, 4, 5 where k is any number

From the equation

Z² = X² + y²

We know that we have a cone, or at least a portion of a cone.  Since we know that square roots will only return positive numbers, it looks like we’ve only got the upper half of a cone.

 So the contour lines in an area are drawn keeping difference in elevation of between two consecutive lines constant.

Z = (x² +y^2)1/2

 

 

Dreamy Draw Recreation Area

Dreamy Draw Recreation Area

Week 11

The Fall 2014 S-STEM Field Trip was to the Dreamy Draw Recreation Area which is the gateway into diverse desert landscapes that offer everything from fantastic views to secluded valleys. We went to the Peak Summit Trail to enjoy the area -- the area boasts dozens of miles of trails to enjoy the glory of the Sonoran Desert in relative solitude.

While I'm looking at those mountains, I tried to calculate the directional derivatives and gradients by using mathematics theories.

First, how to find ‘The directional derivative”:

If the function f(x,y) be the height of a mountain range at each point x=(x,y) and I stand at some point x=a, the slope of the ground in front of me will depend on the direction I am facing. It might slope steeply up in one direction, be relatively flat in another direction, and slope steeply down in yet another direction. The partial derivatives of f will give the slope ∂f∂x in the positive x direction and the slope ∂f∂y in the positive y direction. We can generalize the partial derivatives to calculate the slope in any direction. The result is called the directional derivative.

 

Second, how to find “The gradient”:

There is one direction where the directional derivative Duf(a) is the largest. This is the “uphill” direction. Let's call this direction of maximal slope m. Both the direction m and the maximal directional derivative Dmf(a) are captured by something called the gradient of f and denoted by ∇f(a). The gradient is a vector that points in the direction of m and whose magnitude is Dmf(a). In math, we can write this as

 ∇f(a)∥∇f(a)∥=m and ∥∇f(a)∥=Dmf(a).

 

Newton's cannonball and Gravity

Newton's cannonball and Gravity

Week 10

in this week we will see when the curvature of the Earth is taken into consideration, the direction of gravity changes with the distance traveled. 

 

 Isaac Newton in 1687 in 1687 imagined shooting a cannonball parallel to the Earth's surface from the top of a very high mountain. It would strike the ground at some distance from the mountain top, depending on the velocity of the cannonball, fly off into space or go into orbit around the Earth.

If a cannonball is fired parallel to the Earth's surface from a cannon on the top of a high mountain, the ball will usually travel for some distance until it hits the ground.

If the velocity of the cannonball is only sufficient to carry it several kilometers or miles, the curvature of Earth does not really come into play. For greater distances, the curvature of the Earth becomes a factor.

When the ground is considered flat, the force of gravity is perpendicular to the horizontal velocity of the cannonball. However, when the curvature of the Earth is taken into consideration, the direction of gravity changes with the distance traveled. It is assumed that the force of gravity is concentrated at the center of the Earth.

 

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.