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# ps1 - P207 Fall 2007 Solutions Assignment 1 1 The apparent...

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P207 - Fall 2007 Solutions Assignment 1 1 The apparent size of the elevator is proportional to the angle subtended θ . We see from the diagram that tan θ = h/d where h is the height of the elevator and d is the distance. Since the height is always much h d θ less than the distance, we can use the small angle approximation for the tangent and write that θ h d (a) The angular size of elevator at various distances is given in the table distance angular size angular size (small angle approximation) (exact) d[km] θ [rad] h/d θ [rad] = tan - 1 ( h/d ) 8 5 × 10 - 3 5 . 000 × 10 - 3 4 1 × 10 - 2 1 . 000 × 10 - 2 2 2 × 10 - 2 2 . 000 × 10 - 2 1 4 × 10 - 2 3 . 998 × 10 - 2 0.5 8 × 10 - 2 7 . 983 × 10 - 2 0.25 1 . 6 × 10 - 1 1 . 587 × 10 - 1 1

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(b) Angular size of 40m grain elevator vs distance. 0 0.05 0.1 0.15 0.2 0 2000 4000 6000 8000 10000 θ [rad] distance [m] h/d 2
(c) Log plot of angular size of 40m grain elevator vs distance. The log 0.001 0.01 0.1 1 10 0 2000 4000 6000 8000 10000 θ [rad] distance [m] h/d plot is very nearly a straight line over distances large compared to the height of the object. Therefore, the angular size is an approximately exponential function of the distance.

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ps1 - P207 Fall 2007 Solutions Assignment 1 1 The apparent...

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