Final Project

Alicia D.

Legislator:  William Callegari, Representative

(Click on the image above to enlarge.)

Math Nerd Goes to Mars

They say that math is the universal language. It is the backbone of the universe and the core of NASA. Without it mankind would never have touched the moon or floated in the deep eternity of space. Throughout this program we have been dreaming up ways to travel to Mars and survive on its treacherous surface, which is why I chose math for my final project. In unit eleven we were given six problems to do for a project and here they are.

1.) Compare the Pathfinder velocity at a with earth’s orbital velocity at a.

What is the difference and why?

To do this problem I used my answer from unit eleven and I used the equation

v={2(K+GMm/r)/m}1/2. I then had to find the gravitational constant and the mass of the Earth (5.9736 e24).

The Earth’s velocity at point a equals 2927131849 meters per second.

The Pathfinder’s velocity at point a equals 32140.0444 meters per second.

The difference is 2927099708.9556 meters per second. This difference in velocity is so great because the difference in mass is so great. The vis-viva equation shows an increase in velocity directly with an increase in mass. This is because the orbital energy is made of constants and mass.

2.) Compare the Pathfinder velocity at b with mars’ orbital velocity at b.

Again, what is the difference and why?

To do this problem I had to find the mass of Mars (6.421e23) and use the equation again.

Mars’ velocity at point b equals 1908999091 meters per second.

The Pathfinder’s velocity at point b equals 20960.8985 meters per second.

The difference is 1908978130 meters per second. The reason for this difference is the same as before. A change in mass changes the velocity.

3.) The earth has a radius of 6400 km and spins once on its axis in 24 hours.

Calculate the velocity of a point at the equator in km/sec and mph.

I found the circumference of the earth with the equation C= 2r.

Then I divided it by the number of seconds in a day (86400), and converted to miles per hour.

The velocity of a point is .4654 km/sec, or 1005.3096 mph.

4.) When viewed from celestial north, the Earth both rotates and revolves counter-clockwise. Do the orbital and rotational velocities add or subtract at local midnight? How about at local noon? What considerations might affect the time of day for a launch? Why did NASA launch the Pathfinder spacecraft eastward?

The time of day affects when the Pathfinder was launched because there is a certain window that must be obtained. The Pathfinder was launched eastward so that it could follow the orbit of Mars, so that it could catch up with its destination with minimal difficulties.

5.) In spacecraft design, energy is sometimes expressed in terms of change in velocity required to achieve orbit (“delta-vee” or Dv). Given what we’ve just done, what Dv does the Pathfinder require at a?

This answer is simply the difference between the Pathfinder’s velocity at point a and the velocity at point b.

The Pathfinder requires 11179.1459 meters per second.

6.) Actually, additional energy (velocity) is required for a spacecraft just to escape the Earth’s gravitational field. This velocity is given by the expression vEscape = (2GMEarth/rEarth)^1/2. With MEarth = 6 X 10^24 kg, calculate this velocity in km/sec and mph. This velocity must be added to the Dv calculated in Problem 5. How much, as a percent, does the result change compared to the value obtained in Problem 5? Does leaving the Earth gravity well cost a lot in fuel?

First, I put all the necessary numbers into the equation and added that to the velocity in question five. I used rEarth as the radius of Earth given in problem three. Then I found the percent.

Dv equals 11.3729 kilometers per second or 24565.4231 mph

The result changed 1.733 %.

Sources:

Mars Team Online

http://quest.arc.nasa.gov/mars/ask/path-cruise/

From this source I found information about the orbit of Mars and Pathfinder.

Astrodynamic Constants and Parameters

http://ssd.jpl.nasa.gov/astro_constants.html

This website gave me the numbers used in the equations.

Mars Statistics

http://www.solarviews.com/eng/mars.htm#stats

This site gave me the mass of Mars for the vis-viva equation.

Last Updated:  09/10/01