Estimating a tower's height using shadows

[jachyra]

Hi all!

I was reading the following article on howstuffworks.com:
http://science.howstuffworks.com/question379.htm

It says to apply the following procedure to estimate the building height:
1. Measure the length of the broomstick's shadow
2. Calculate the ratio of the broomstick's shadow length to the broomstick's height
3. Measure the tower's shadow
4. Apply the ratio to discover the tower's height.

Attached is a picture I drew showing why this simple calculation seems confusing to me. The black box represents the stick placed beside the building (white box). If the rays of the suns strike the ground at different angles, then how can a similar triangle approach using just ratios of the lengths be used to calculate the building height?

Is my drawing wrong? Have I assumed something wrong by drawing the sun as a point of light?

 

[Shooting Star]

I think so. The sun's rays are parallel, remember, because the sun is so far away. Now try drawing the same diagram with the direction of sunlight same everywhere. Immediately you get similar triangles.

 

[Rib5]

Thats one thing that sometimes confuses me about physics, that you have to make assumptions or approximations sometimes to get better results.

This kind of thing happens in optics when you have lenses like in telescopes, you just assume that the stuff is so far away that the light coming from it is parallel.

 

[Shooting Star]

It's not so much of an assumption as a matter of practical convenience. Can you think of a simple experiment to show that the sun's rays are not parallel? (In reality, they are not exactly parallel.) For stars, they are parallel for almost all practical purposes.

 

[stewartcs]

Is my drawing wrong? Have I assumed something wrong by drawing the sun as a point of light?

Yes your drawing is wrong. The reason the ratio method works is because the two objects form similar triangles. Like Shooting Star mentioned, the hypotenuse of the triangles (sun's rays) are assumed to be parallel which gives the similar triangles.

This may help explain a little better...

http://mathforum.org/library/drmath/view/55238.html

 

[DaveC426913]

It's not so much of an assumption as a matter of practical convenience. Can you think of a simple experiment to show that the sun's rays are not parallel? (In reality, they are not exactly parallel.) For stars, they are parallel for almost all practical purposes.

You're right they're not. Two rays 20 feet apart diverge by 20 feet over 93 million miles or about 1 part in 24 billion, or 7 x 10^-13 degrees.

So, unless your estimation of the building height needs to be accurate to 13 decimal places, you'll be OK.




Actually, since we're getting nitpicky, we have to aco**** (that was supposed to be "account") for the fact that the sun is not a point source of light. Its rays come from a disc, which causes soft-edged shadows. This effect far overwhelms the divergence of the rays - actually there are more converging rays than there are diverging...

 

[topherfox]

This effect far overwhelms the divergence of the rays - actually there are more converging rays than there are diverging...

Not the mention trying the measure the shadow! The instrumental error involved :P

 

[jachyra]

ohhhhhhhhhhhhhh!

It's all clear now. Thanks for all the information! I just joined this forum yesterday and I already love it! You guys rock.

 

[Shooting Star]

Does it mean something?

... we have to aco**** (that was supposed to be "account")...

Why? I mean why aco**** and then the explanation in the bracket?

 

[DaveC426913]

Why? I mean why aco**** and then the explanation in the bracket?

I typed my message, then hit preview, and this is what it did to my (obviously mistyped) message.

"Hah, what a silly bunt I am."

 

[Shooting Star]

I typed my message, then hit preview, and thi sis what it did to my (obviously mistyped) message.

"Hah, what a silly bunt I am."

No need to go public on that. (The mistyping, I mean, of course )