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Unformatted text preview: P207  Fall 2008 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 (b) Angular size of 40m grain elevator vs distance. 0.05 0.1 0.15 0.2 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 0.001 0.01 0.1 1 10 2000 4000 6000 8000 10000 θ [rad] distance [m] h/d plot of log( θ ) vs distance is an approximately straight line for distances much greater than the height of the object (say beyond 4000m), suggesting that in that regime, the angular size is very approximately an exponential function of the distance. Of course we know that when h/d 1 that the angular size θ ∼ h/d . Then log( θ ) = log( h ) log( d ) If we were to make a loglog plot, that is plot log( θ ) vs log( d ) then we would get a straight line with slope 1....
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 Fall '07
 LIEPE, M
 Physics, Mass, Velocity, Angular Size, Cayuga Lake

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