Question

MISSION TO MARS
Prepping for Launch
200 years has passed and now it is year 2220. The Earth is out of basic resources as they have been drastically drained in the past 200 years. The president of the United World Council (UWC) has approved you and your crew’s Mission to Mars. You will pilot the most advanced spaceship the world has ever known, the Excelsior! It will carry equipment that will help to transform Mars so as to resemble the Earth, especially so that it can support human life.
Before the mission can launch a few items need to be figured out. What is the capacity of the fuel tank? How long should the fuel burn to achieve escape velocity (otherwise the Excelsior will be stuck in the Earth’s gravitational pull)? How long will it take the Excelsior to arrive at Mars?
1. The ground crew is filling the fuel tanks. You know they fill the tanks at a rate of r=100����, where r is measured in tons per hour and t is measured in hours. They fill the tanks for two and a half hours. Calculate how many tons of fuel are in the tanks once they are full, rounded to the nearest ton. (Hint: Use Integration)
2. You know the Excelsior must achieve 30,000 mph to reach escape velocity. The rocket’s engines are state-of-the-art and capable of creating an acceleration based on the profile a=10��, where t is in seconds and a is in ����/��2. Unfortunately the pull of gravity works against the rocket’s motion at 32 ����/��2. Neglecting wind
resistance, calculate how long to burn the fuel to allow the Excelsior to achieve 30,000 mph, rounded to the nearest minute. (Hint: Don’t forget to convert mph to ����/��2)
3. The straight-line distance from Earth to Mars at the time the launch is scheduled is 100 million miles. In order to avoid other celestial bodies the Excelsior must travel in an arc given by the equation ��= −1125(��−50)2+20, where x and y are in millions of miles, and the x-axis denotes the straight-line distance to Mars. Find the actual distance travelled by the Excelsior rounded to the nearest million and then find the approximate time for the trip rounded to the nearest month, assuming a constant speed of 30,000 mph and 30 days in each month. (Hint: Use Arc length formula =
��
∫ √1 + (����

��
����)
2 dx )

4. The trajectory of the spaceship Excelsior during the first 5 minutes of the launch can be represented by an equation for its altitude
H(T) = 2008 - 0.047 T3 + 18.3 T2 – 345T
and an equation for its downrange distance due-east
R(T) = 4680 e 0.029T
where the distances are provided in units of feet.
a) Use the parametric equations for h(T) and R(T) to determine the equation for the speed, S, of the Excelsior along its trajectory where
����
����= ( (����
����)2 + (����
����)2)1/2.
b) Determine the formula for the magnitude of the acceleration of the spaceship Excelsior using the second time derivatives of the parametric equations.
5. Write code for part 1 and part 2 in any programming language. (Note: Code should be generic that is for part 1 it should work for any value of t and for part 2 it should work for any value i.e. not only for 30,000 mph. Include the screenshots of your code in project document and also submit code file).
Good luck

Answer 1

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