Question

A laser beam is to be directed toward the center of the moon, but the beam strays 0.5 degree from its intended path.

a. How far has the beam diverged from its assigned target when it reachcs the moon? (The distance from the earth to the moon is 240,000 mi.) b. The radius of the moon is about 1000 mi. Will the beam strike the moon?

Answer 1

The laser beam, straying 0.5 degrees from its path to the Moon, diverges approximately 2,088 miles from its target. Since this is greater than the Moon's radius of about 1,000 miles, the laser beam will miss the Moon.

Step-by-step explanation:

The student's question involves calculating the divergence of a laser beam that strays 0.5 degree from its path as it travels to the moon and determining if it will strike the moon. Given that the distance from the Earth to the Moon is 240,000 miles and the moon's radius is about 1,000 miles, we can solve this using trigonometry.

First, we convert the angle to radians: 0.5 degrees * (π / 180) = approximately 0.0087 radians. Then, we calculate the linear distance of the divergence (D = angle in radians * distance to Moon).

D = 0.0087 * 240,000 miles
= approximately 2,088 miles. The laser beam diverges approximately 2,088 miles from its target.

Considering the radius of the Moon is about 1,000 miles, the laser beam would miss the Moon, since the divergence distance is greater than the Moon's radius.

Answer 2

a. The laser beam diverges approximately 2093.76 miles from its intended target on the moon. b. Yes, the beam will strike the moon as it is well within the moon's radius.

Step-by-step explanation:

a. To calculate how far the laser beam has diverged, we can use the formula for the length of an arc on a circle. The distance diverged d is given by
`\(d = (\theta)/(360^\circ) * 2\pi r\)`, where r is the distance from the Earth to the moon (240,000 miles), and
`\(\theta\)`is the angle of divergence (0.5 degrees). Substituting these values, we find
`\(d \approx (0.5)/(360) * 2\pi * 240,000 \approx 2093.76\) miles.`

b. The radius of the moon is about 1000 miles. Since the laser beam's divergence (2093.76 miles) is greater than the radius of the moon, the beam will indeed strike the moon. The laser beam covers an area larger than the moon's radius, making contact with the lunar surface.

In conclusion, despite straying 0.5 degrees from its intended path, the laser beam diverges approximately 2093.76 miles but still manages to strike the moon, which has a radius of about 1000 miles. These calculations provide insights into the precision required for such astronomical endeavors and the tolerances within which they can still be successful.