Lecture 6 Parametric curve Rate of change of position with time In terms of curve Example Helix Rules for vector derivatives r cost sint t v sint cost 1 v 2 a cost sint 0 Suppose u is a function v and w are vector functions a R Linear Product Example magni cation u t 1 t magni cation at time t Circle in plane r t cost sint What I see u t r t u t r t 1 t cost sint Curve cost sint veloctiy sint cost speed 1 length speed 1 2 Curve acost bsint velocity asint bcost a sin t b cos t length Kepler s Laws describes planetary motion 1 Orbit is in a plane and is ellipse with foci at sun 2 The line from the sun to the plant sweeps out area at a constant rate 3 Relates the period of revolution to the major semi axis Theorem The second law implies that the acceleration points towards the sun Prove using vector calculations r t vector from the sun to the planet From our computations we see that the second law implies that r t r t is constant Claim r t r t is constant We must show that the direction is constant The rst law implies that r and r live in a plane therefore r r is perpendicular to this plane Vector constant shows that direction is zero
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