lecture6-09[1]

lecture6-09[1] - Math 18.02 (Spring 2009): Lecture 6...

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Math 18.02 (Spring 2009): Lecture 6 February 13 Reading Material: From Simmons : 17.4 and Lecture Notes K. Last time: Parametric equations for lines and curves Today: · , × . Kepler’s second law. 2 Velocity and Acceleration We recall that a curve γ in 3D is described by three parametric equations x = x ( t ) y = y ( t ) z = z ( t ) where t belongs to an interval [ a, b ]. We can associate to the point P ( t ) = ( x ( t ) , y ( t ) , z ( t )) that determines the curve the vector position --→ OP ( t ) = ~ r ( t ) = x ( t ) ˆ i + y ( t ) ˆ j + z ( t ) ˆ k, that is the vector that starts at the origin O and ends on the point P ( t ) of the curve: It is now clear that in order to describe a curve it is completely equivalent to either give parametric equations or give the vector position ~ r ( t ). Definition 1. Given a curve γ described by a vector ~ r ( t ) = x ( t ) ˆ i + y ( t ) ˆ j + z ( t ) ˆ k for t in [ a, b ] , we define the velocity of ~ r ( t ) to be the vector ~v ( t ) defined as ~v ( t ) = dx dt ( t ) ˆ i + dy dt ( t ) ˆ j + dz dt ( t ) ˆ k = x 0 ( t ) ˆ i + y 0 ( t ) ˆ j + z 0 ( t ) ˆ k. 1
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We define the speed of ~ r ( t ) to be s ( t ) = | ~v ( t ) | = p ( x 0 ( t )) 2 + ( y 0 ( t )) 2 + ( z 0 ( t )) 2 . We usually also write ~v ( t ) = d~ r dt ( t ) , or even simpler ~v = d~ r dt . The vector
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lecture6-09[1] - Math 18.02 (Spring 2009): Lecture 6...

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