Calculating Distance in Circular Motion: Earth to Moon Spacecraft Force Analysis

AI Thread Summary
To determine the distance from the center of the Earth where a spacecraft experiences a net force of zero while traveling to the moon, one must analyze the gravitational forces exerted by both the Earth and the Moon. The gravitational force on the spacecraft can be expressed as the sum of the forces from both celestial bodies, represented by the equation F=F_earth+F_moon. By setting the forces equal to zero, the spacecraft's position can be calculated along the axis between the Earth and the Moon. This involves solving the equation -G(M_earth m/x^2) + G(M_moon m/(R-x)^2) = 0. Understanding this balance of forces is crucial for spacecraft trajectory planning.
futb0l
A spacecraft leaves Earth to travel to the moon. How far from the centre of the Earth is the spacecraft when it experiences a net force of zero?

Can anyone help me with this?

Thanks.
 
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Draw an axis. Place the centre of the Earth at x=0 and the centre of the moon at x=R.
Then the gravitational force on a spacecraft of mass m at x between 0 and R is:

F=F_{earth}+F_{moon}=-G\frac{M_{earth}m}{x^2}+G\frac{M_{moon}m}{(R-x)^2}
 
Thanks for that :)
 

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