Name: halp me! 2010-05-23 14:49
Yo guys. I'm stumped with this math problem I have to turn in tonight. Can any of you guys kick me in the right direction or help me.
Mars Math - Part Three
Calculate 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.
To perform this calculation engineers use an equation that gives the velocity of an object at various points on an elliptical orbit called the vis-viva equation. This equation was determined by the German scientist Gottfried Leibniz in the 17th century. 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. This equation allows you to calculate the orbital velocity of the spacecraft at the point of departure (Earth’s orbit) and the point of arrival (Mars’ orbit).
The vis-viva equation is derived below by setting kinetic energy + gravitational potential energy equal to an energy constant (K). The constant (K) is calculated from the mass of the primary agent of the gravitational force (the sun, in the case of our solar system) and the semi-major axis of the orbit and is equal to -GMm/2a.
Vis-viva equation derivation:
kinetic + gravitational = energy constant (K)
energy energy
½ mV2 + - GMm/r = -GMm/2a
V2 + - 2GM/r = -GM/a (Multiplying by 2 and dividing by m)
V2 = 2GM/r + -GM/a (Rearranging)
V = {2GM (1/r - 1/[2a])} 1/2
V = (2GM)1/2 ∙ (1/r - 1/[2a]) 1/2
V = 1.6 X 1010 (1/r - 1/[2a]) 1/2 ( GIVEN (2GM )½ =1.6 X 1010 )
The terms used in the vis-viva equation are familiar by now and are defined below:
K = energy constant = -GMm/2a
V = Orbital Velocity in m/s
M = Mass of the primary object (sun) in kilograms
m = mass of the secondary object (spacecraft) in kilograms
G = 6.67 X 10-11 N∙m2/kg2 (gravitational constant)
r = the distance from the Earth (for Point A), or Mars (for Point B), to the sun in meters.
a = semi-major axis of the Hohmann Transfer ellipse in meters (use answer from module 9)
Use http://nssdc.gsfc.nasa.gov/planetary/factsheet/marsfact.html for values of r at aphelion.
Remember: Both terms, r and a, need to be converted to meters.
Your final answer needs to be in miles per hour so you will need to convert your answer from meters/second to kilometers/second, and then to miles per hour.
***Module 9's answer is 1.847x10^8
Mars Math - Part Three
Calculate 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.
To perform this calculation engineers use an equation that gives the velocity of an object at various points on an elliptical orbit called the vis-viva equation. This equation was determined by the German scientist Gottfried Leibniz in the 17th century. 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. This equation allows you to calculate the orbital velocity of the spacecraft at the point of departure (Earth’s orbit) and the point of arrival (Mars’ orbit).
The vis-viva equation is derived below by setting kinetic energy + gravitational potential energy equal to an energy constant (K). The constant (K) is calculated from the mass of the primary agent of the gravitational force (the sun, in the case of our solar system) and the semi-major axis of the orbit and is equal to -GMm/2a.
Vis-viva equation derivation:
kinetic + gravitational = energy constant (K)
energy energy
½ mV2 + - GMm/r = -GMm/2a
V2 + - 2GM/r = -GM/a (Multiplying by 2 and dividing by m)
V2 = 2GM/r + -GM/a (Rearranging)
V = {2GM (1/r - 1/[2a])} 1/2
V = (2GM)1/2 ∙ (1/r - 1/[2a]) 1/2
V = 1.6 X 1010 (1/r - 1/[2a]) 1/2 ( GIVEN (2GM )½ =1.6 X 1010 )
The terms used in the vis-viva equation are familiar by now and are defined below:
K = energy constant = -GMm/2a
V = Orbital Velocity in m/s
M = Mass of the primary object (sun) in kilograms
m = mass of the secondary object (spacecraft) in kilograms
G = 6.67 X 10-11 N∙m2/kg2 (gravitational constant)
r = the distance from the Earth (for Point A), or Mars (for Point B), to the sun in meters.
a = semi-major axis of the Hohmann Transfer ellipse in meters (use answer from module 9)
Use http://nssdc.gsfc.nasa.gov/planetary/factsheet/marsfact.html for values of r at aphelion.
Remember: Both terms, r and a, need to be converted to meters.
Your final answer needs to be in miles per hour so you will need to convert your answer from meters/second to kilometers/second, and then to miles per hour.
***Module 9's answer is 1.847x10^8