Spot Size on Moon from Earth Laser Beam

[nannygreen]

I need some help on how to start this problem and I think I can work it out.
A laser beam is directed at the moon 380,000 km from earth. The beam diverges at an angle of 1.8 x 10-5 rad. How large a spot will it make on the moon?

 

[BobG]

That would be you equation for finding the length of an arc.

$$s=r\theta$$

Technically, that gives you the diameter on a flat moon.

You'd then have to figure out how big of an angle that is based on the Moon's radius and use your equation of the length of an arc once again. Considering the size of your spot as compared to the radius of the Moon (1738.1 km), it's not worth it (which is why the radius of the moon wasn't given). For the number of significant digits given, accounting for the curvature of the Moon winds up being the same as for a flat moon.

 

[wais]

To solve this problem, we can use the formula for angular diameter:

Angular diameter = Spot size / Distance

First, we need to convert the given angle in radians to degrees:
1.8 x 10^-5 rad = 1.8 x 10^-5 x (180/π)° = 0.0010324°

Next, we can plug in the values into the formula:
Angular diameter = Spot size / Distance
0.0010324° = Spot size / 380,000 km

Solving for the spot size, we get:
Spot size = 0.0010324° x 380,000 km = 391.56 km

Therefore, the spot size on the moon from the earth laser beam will be approximately 391.56 km.