Escape Velocity and Satellites

Escape Velocity and Satellites

Week 14

If we ask our self why the moon not fall on the earth? Why the earth and the planets are orbiting the sun, what force is keeping them in orbit?

 

This question was answered by Newton’s Law of Universal Gravitation.

“Any two objects attract each other with a gravitational force, proportional to the product of their masses and inversely proportional to the square of the distance between them.

 

1.      If Earth is a uniform sphere with mass 𝑀 the gravitational force between Earth and the body near the earth is:

𝐹=𝐺 (𝑀𝑚 )/𝑟^2

 

2.      If the body is far from the Earth, the gravitational potential energy is:

𝐹(r) = -𝐺 (𝑀𝑚)/𝑟^2
 

 

3.      If the body’s initial kinetic energy is equal to the potential energy at the Earth’s surface, it means their total energy will be zero. So the velocity is called the Escape Velocity.
 

 

4.   The Satellites are usually put into orbit around the Earth. If the tangential speed high enough, so the satellite put into orbit around the Earth. It does not return to Earth, but not so high that it escapes Earth’s gravity altogether.
 

 

Saddle Point

 

Saddle Point

Week 13

Thanksgiving was yesterday and I ate turkey dinners at home with my family. Today several stores such as Target, Walmart, and Kmart are opening their doors hours during Black Friday. Hundreds shoppers will be able to take advantage of special deals available during this time.

As usual, I am thinking mathematically how can see this procurement. How to minimize the total procurement cost considering discount prices.

 

I can find the saddle point which is a point in the range of a function that is a stationary point but not a local extremum. So, I can find the saddle point for a one-dimensional function, for example:

f(x)=x^3,

 f^'(x) = 3x^2

f^('')(x) = 6x  

f^(''')(x) = 6.

This function has a saddle point at x₀=0 by the extremum test since:

f^('')(x₀)=0 and f^(''')(x₀)=6≠0.

In one dimension, a saddle point is a point which is both a stationary point and a point of inflection. Since it is a point of inflection, it is not a local extremum.

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.