Chapter 6.2, Problem 49E

Deviation of a Laser Beam A laser beam is to be directed toward the center of the moon, but the beam strays 0.5° from its intended path.

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

To determine

To find: The distance of the beam diverse from its assigned target to reach the moon.

Answer to Problem 49E

Distance of the beam diverse from its target to reach the moon is $2100\text{mi}$.

Explanation of Solution

Given:

The direction of the laser beam is towards the center of the moon.

Distance of earth and moon is $240,000\text{mi}$.

Consider the above figure (1) to calculate the value of $d$.

Formula Used:

Trigonometry property of tangent is $\text{tan}\theta =\left(\frac{\text{opposite}}{\text{adjacent}}\right)$.

Calculation:

The below figure represents the direction of laser beam towards moon.

Let $d$ is the distance between the beam and moon.

The value of d is calculated as follows,

Substitute $\theta =0.5°$ and 240,000 for adjacent and $d$ for opposite in $\text{tan}\theta =\left(\frac{\text{opposite}}{\text{adjacent}}\right)$,

$\begin{array}{c}\text{tan 0}\text{.5}°=\left(\frac{d}{240,000}\right)\\ d=\left(\text{tan}0.5°\times 240,000\right)\\ =\left(0.5463\times 240,000\right)\left(\because \tan 0.5°=0.\text{5463}\right)\\ d\approx 2100\text{mi}\end{array}$

Therefore, the distance of the beam diverse from its assigned target to reach the moon is $2100\text{mi}$.

To determine

To find: Whether the beam strike the moon or not.

Answer to Problem 49E

The beam will not strike the moon.

Explanation of Solution

Calculation:

The direction of the laser beam is towards the center of the moon.

Distance of earth and moon is $240,000\text{mi}$.

Radius of moon is $1000\text{mi}$.

And from part (a) the distance of the beam diverse from its assigned target to reach the moon is $2100\text{mi}$.

The radius of moon is $1000\text{mi}$ and distance from the earth is $2100\text{mi}$.

Then, the beam will not strike the moon.