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Finding an Expression for the Universal Gravitational Potential Energy — with Calculus and Python
Here is a physics problem. You have an object above the moon and you drop it. What is the final speed before the object hits the surface? Note: this object is NOT the lunar lander since it also has a horizontal motion — but you get the idea. The best way to solve this problem is to use the gravitational potential energy (with respect to infinity) of the object-moon system. This potential looks like this.
In this expression, r is the distance from the center of the moon to the object and G is the universal gravitational constant (6.67 x 10^-11 N*m²/kg²). But how do you get this expression? I’m going to show you.
Let me start with the work-energy principle. This says that the work done on a system is equal to its change in energy. As an equation, it just looks like this.
If you have a constant force applied to an object over a displacement in a straight line, then the work would be defined as:
Where this is the dot product of two vectors. However, if the force is NOT constant or the displacement is not in a straight line, then you would need to calculate the work as a line integral.
Line integrals can get tricky if you aren’t moving in a straight line, so we will just keep it…
