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Unformatted text preview: Homework # 5 Solutions 1. In example given in the notes describing the observations of col- lapsing clouds, it is indicated in some cases that there is a locus of points for which v r is constant. v r is the radial velocity seen by an observer in the z direction (see the relevant figure). For the case that v ( r ) = v r 1 , find equations for ( r, ), or ( x, y ) where r = radicalbig x 2 + y 2 and = tan 1 ( x/y ), of the locus of points for which v r = 2 v /R i = constant. Here, R i is the outer radius of the collapsing region. Using v ( r ) = v /r , and defining the angle relative to the axis perpindicular to the line-of-sight, a point along the line-of-sight R which is a distance z from the axis satisfies sin = z/r . Then the component of the velocity along the line-of-sight v r = v ( r ) sin = ( v /r ) sin . Solving for r : r = ( v /v r ) sin = ( R i / 2) sin ....
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- Spring '11