can i tell how far up in the sky a 1m dia object is by its apparent size.if it went straight up, and its last sighting it appeared to be 1mm in diameter, what is its height?
An object has diameter of 1m; it appears to have a diameter of 1mm. How far away is it?
(1) It is far more convenient (and accurate) to discuss the apparent size in terms of an angle. When you describe a cloud as having the same apparent size as a tennis ball, the listener does not know if you mean a tennis ball held close to the face, at arm's length, 10m away, etc...
(2) I will assume that when you say the apparent size is 1mm, you measured this with a ruler held at arm's length. I assume the ruler is 60cm from your eye. Then the angle in the sky can be found:
** Draw an isosceles triangle; label the vertex angle as `alpha` , the altitude as d (for distance), and the base as w (for actual width). Then from trigonometry we have `tan(alpha/2)=(w/2)/d` . Then `d=1/(2tan(alpha/2))` . Also `alpha=2tan^(-1)(w/(2d))` **
The angle subtended by a 1mm object at .6m is `alpha=2tan^(-1)(.001/1.2)~~.095` , so your object covers about `.1^circ` of the sky. (Under the assumptions I made)
(3) Then the distance of the 1m object is `d=1/(2tan(alpha/2))=1/(2tan.05)~~572.96`
Thus the object was approximately 573m in the air.
Again, a better approach is to measure the angle, as this is more consistent.