Motion under a central force
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Kepler's Second Law
Kepler's second law states that the line joining an object in orbit to the sun sweeps out equal areas in equal times. [Diagram goes here - download the original to see it.] If the area X is equal to the area Y then the object in orbit takes the same time to travel from A to B as it does to travel from C to D. Kepler's second law can be derived from the fact that for a body subject only to a central force [Equation goes here - download the original to see it.] Consider a sector of an ellipse [Diagram goes here - download the original to see it.] In time the object P travels from [Equation goes here - download the original to see it.] The area of the sector swept out in this time can be approximated by a triangle [Diagram goes here - download the original to see it.] The area of the triangle is [Equation goes here - download the original to see it.] This is an application of the usual result for a triangle with sides a,b and included angle that its area is [Equation goes here - download the original to see it.] The rate of change of the area is [Equation goes here - download the original to see it.] As [Equation goes here - download the original to see it.] the approximation becomes exact; furthermore, [Equation goes here - download the original to see it.] So the rate of change of the area is [Equation goes here - download the original to see it.]Since [Equation goes here - download the original to see it.], the rate of change is constant. This means that the area swept out by a radical line in any constant period of time must be constant.
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Contents of
Motion under a central force
1 Motion under a central force
2 Transverse Velocity
3 Kepler's Second Law
4 Motion under gravity
5 Energy and central force systems
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