| Mission:
Possible
MISSION |
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| Your assignment
is to design one of the elements needed for the first human trip to Mars
and to submit a drawing and a ½ page description of how it works,
and the Mars Math question (optional on this assignment) via the Comm
Link.
Click here to complete the Quick Quiz! for this lesson. Click on Extended Mission for more fun activities, links, and resources on this topic. Consider this topic for your final project! Assignment – Mission Possible Design one of the elements needed for the first human trip to Mars for your assignment. Choose either a piece of equipment, an apparatus, or a facility that will be needed by future explorers and submit a drawing and a ½ page description of how it would be constructed, it's structure, and what it would be used for (form and function). Possible topics include:
Mars Math (optional) Part Three (Continued from Lessons Nine and Ten) We need to determine the orbital velocity of our Mars spacecraft at the point of departure, when it leaves Earth orbit, and at the point of arrival at Mars. We can do this using an equation that gives the velocity of an object at various points on an elliptical orbit. We will use the vis-viva equation which was determined by the German scientist Gottfried Leibniz in the 17th century. This will allow you to predict the orbital velocity of the spacecraft at the point of departure (Point A on the diagram) and the point of arrival (Point B on the diagram). You will need to use your answer from the last Mars Math problem for the semi-major axis to complete this equation. The term, "vis viva," derives from the Latin, vis = force or power, and viva = living. In the older writings, it was associated with the ability of a body to do work on its environment. Now it usually refers to the principle of energy conservation. One derives the velocity of a planet or spacecraft in its orbit by writing out the vis viva equation: kinetic energy + gravitational potential energy = a constant (K) The constant is calculated from the mass of the primary (the Sun, in the case of our solar system) and the semi-major axis of the orbit. The kinetic energy term contains the square of the velocity. The vis-viva Equation is below (please use the bottom version for your problem): K= energy constant
v= {2(K+GMm/r)/m} ^1/2 K= -GMm/2a Therefore, v= {2GM (1r – 1/[2a])} ^1/2 Or, finally (for your use) v= 1.6 x 10^10 (1/r – 1/[2a]) ^1/2 Use the equation above to determine the velocity at point A (departure) and at point B (arrival) on the diagram below in kilometers/sec and in miles per hour. Your units are meters/second. (If you trace the sequence of the three equations and balance them you will see how this occurs). Remember: · r and a need to be
converted to meters...you must use exponents!
1. The distance from
the Earth to the Sun is 150 million km
r = the distance from the Earth (for Point A), or Mars (for Point B), to the Sun a – semi-major axis of the ellipse v= 1.6 x 10^10 (1/r – 1/[2a])^1/2 Clues: (Only if you need them!) 1 million kilometers can be expressed as 1(10^9) v= 1.6 x 10^10 (1/r – 1/[2a])^1/2 can be expressed as v = 1.6 x 10^10 {SQUARE ROOT (1/r – 1/[2a])}
http://www.treasure-troves.com/bios/Leibniz.html
Thank you to Joe Kolecki and NASA’s Learning Technology Project at the Glenn Research Center for use of these questions and activities. Below are some more math activities on this topic you can consider for your final project if math is your specialty! 1. Compare the Pathfinder velocity at a with
earth’s orbital velocity at a. What is the difference and why? (Express
your answer in terms of total orbital energy.)
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. 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? 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? 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
Your assignment will be assessed using
the following criteria. If your work meets all of the criteria below it
will be considered exemplary. If one or two of the criteria are not met
your work will be considered satisfactory. If more than three criteria
are not met your work will be considered unsatisfactory and you will be
asked to resubmit.
Don’t forget you need to complete the quiz before you go on to the next lesson. Click here to complete the Quick Quiz! |
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