Acceleration with respect to displacement?

AI Thread Summary
The discussion centers on the physics of launching a rocket from the moon's surface during the Northrop Grumman Lunar Lander Challenge. The key question involves understanding how the force required to maintain altitude on Earth compares to that on the moon, considering the differing gravitational forces of 9.8 m/s² on Earth and 1.62 m/s² on the moon. As the rocket ascends, the gravitational force diminishes with distance, leading to increased acceleration. The user seeks assistance in calculating the rocket's velocity and displacement after 180 seconds of constant upward force. The conversation highlights the complexities of gravitational effects on rocket dynamics in a lunar environment.
DavyCrocket20
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Hi. My name is David. I like physics. Recently I ran into a problem that has really begun to bug me. Have you heard of the Northrop Grumman Lunar Lander Challenge? It's really cool. The teams are competing for a purse of like $2 million. For the highest level, the teams must launch their rocket from point A and remain in the air at an altitude of at least 50 meters for 180 seconds. Then they land their rocket at point B. After measuring the accuracy of the landing, the rocket makes a return flight back to point A, again at an altitude at or exceeding 50 meters for 180 seconds. Cool Challenge.

My question relates to how the same force required to suspend a mass for 180 seconds on Earth would affect the same mass but starting from rest on the surface of the moon. At first, the same force will be enough to produce an acceleration of 9.8m/(s*s)(earth gravity)-1.62m/(s*s)(moon gravity). I am assuming the rocket is launched vertically away from the moon's surface. The rotation of the moon can be ignored as it is a mere ~10 miles/hour. The difficulty arises as the rocket gains velocity and moves away from the moon. The force of gravity exerted by the moon diminishes as a function of distance. This in turn causes an increase in the acceleration of the craft. This of course relates directly to the distance of the object from the moon. I seem to be caught in a circular loop of dependent factors. Can anyone help me? This isn't homework so don't feel pressured. I just really want to figure this out.

The force of gravity between two objects (aka a rocket and the moon) is equal to G*m1*m2/(r*r) where G is the gravitational constant: 6.67300*10^(-11)*m^(3)/(kg*s*s) and m1 and m2 are the masses of the objects in question and r represents the distance between them. The moon's mass is 7.36*10^(22)*kg and its radius is 1 737.4*km.

Thanks in advance to any kind genius that would like to help me!
 
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Here, I'll do my best to sum up my problem mathematically:
Find the velocity and displacement of a rocket taking off of the moon at t=180. The rocket's engine supplies a constant force upward (directly opposite to the force of gravity) of RocketMass*9.81m/s/s.
 
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