Angular size in degrees problem,

[astru025]

Homework Statement

For distant objects, the angular size in degrees can be approximated as 57° (w/d), where w is the width of the object and d is its distance. What is the angular size of the headlights on a car 13 km away if the headlights are 1.5 m apart?

Homework Equations

D/w

The Attempt at a Solution

13 km to m = 13000 m. 1.5 m / 13000 m = 1.15E-4. This answer was incorrect and I'm trying to figure out what I'm doing wrong. Any help would be really nice, thanks!

 

[CWatters]

What does "57° (w/d)" mean?

Are you sure it doesn't say "0.57(w/d)" ?

I ask because the correct angle would be about Cos^-1(w/d) which is 0.66 E-4 and..

0.66/1.15 = 0.57

 

[nasu]

The Attempt at a Solution

13 km to m = 13000 m. 1.5 m / 13000 m = 1.15E-4. This answer was incorrect and I'm trying to figure out what I'm doing wrong. Any help would be really nice, thanks!

The answer is not incorrect but it is in radians.
Your problem "expects" you to calculate in degrees. Why don't you just apply the full formula given in the text? Multiply by 57 degree.
The factor of 57 is the approximate conversion factor between degrees and radians. (57 degrees/radian).

 

[astru025]

Okay thanks I did 57 x (1.5/13000) and got .0066 degrees which was correct. Thanks

 

[Nickie]

The formula you are using (D/w) is actually for calculating the distance of an object based on its angular size and physical size. In this problem, we are trying to find the angular size of the object based on its physical size and distance. The correct formula to use would be w/d, where w is the width of the object and d is the distance.

In this case, the angular size of the headlights on a car 13 km away would be 1.5 m / 13000 m = 0.000115 degrees. This is a very small angle, which is expected for distant objects.

It is important to understand the difference between the two formulas and when to use them. Keep in mind that the D/w formula is used when you know the angular size and want to find the distance, while the w/d formula is used when you know the physical size and distance and want to find the angular size.

I hope this helps clarify the problem for you. Keep practicing and remember to always double check your formulas and units to ensure accurate results. Good luck with your studies!