Estimating a tower's height using shadows

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Discussion Overview

The discussion revolves around estimating the height of a tower using the lengths of shadows cast by a broomstick and the tower itself. Participants explore the assumptions involved in using similar triangles for this estimation, particularly considering the nature of sunlight and its rays.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants question the validity of using similar triangles for height estimation due to the assumption that sunlight can be treated as parallel rays.
  • Others argue that the sun's rays are effectively parallel for practical purposes, given the vast distance from the Earth.
  • A participant mentions that while the rays are not perfectly parallel, the divergence is negligible for most practical height estimations.
  • There is a discussion about the sun being a disc rather than a point source, which affects shadow sharpness and introduces additional considerations for accuracy.
  • Some participants express confusion about the assumptions made in the drawing and the method, indicating a need for clarification.
  • A later reply highlights the potential for instrumental error when measuring shadows, adding another layer of complexity to the estimation process.

Areas of Agreement / Disagreement

Participants generally agree that the sun's rays can be approximated as parallel for the purpose of this estimation, but there is disagreement regarding the implications of this assumption and its impact on accuracy. The discussion remains unresolved regarding the extent to which these factors affect the estimation method.

Contextual Notes

Limitations include the assumption of parallel rays, the effects of the sun's disc shape on shadow sharpness, and the potential for measurement errors. These factors have not been fully resolved in the discussion.

Who May Find This Useful

This discussion may be useful for individuals interested in practical applications of geometry in physics, particularly in estimating heights using indirect measurements and understanding the assumptions involved in such methods.

jachyra
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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?
 

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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.
 
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.
 
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.
 
jachyra said:
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
 
Shooting star said:
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...
 
Last edited:
DaveC426913 said:
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
 
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.
 
Does it mean something?

DaveC426913 said:
... we have to aco**** (that was supposed to be "account")...

Why? I mean why aco**** and then the explanation in the bracket?
 
  • #10
Shooting star said:
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."
 
Last edited:
  • #11
DaveC426913 said:
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 :smile:)
 

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