Problem 49 - Algebra and Trigonometry

Question

Deviation of a Laser Beam A laser beam is to be directed toward the center of the moon, but the beam strays \(0.5^{\circ}\) from its intended path. (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. (b) The radius of the moon is about 1000 mi. Will the beam strike the moon?

Step-by-Step Solution

Answer

The beam diverges about 2094.396 miles; it does not hit the moon.

Step 1: Convert Degrees to Radians

The angle of deviation is given as 0.5 degrees. We need to convert this angle into radians because most trigonometric calculations, such as those needed later, require angles in radians.The conversion formula is:\[ \text{Radians} = \text{Degrees} \times \frac{\pi}{180} \]Substituting the given angle:\[ 0.5^{\circ} \times \frac{\pi}{180} \approx 0.00872665 \text{ radians} \]

Step 2: Calculate Divergence Distance

Using the small angle approximation, the divergence distance can be found using:\[ D = d \times \theta \]Where:- \( D \) is the divergence distance,- \( d \) is the distance to the moon (240,000 miles),- \( \theta \) is the angle in radians (approximately 0.00872665 radians).Substitute the known values:\[ D = 240,000 \times 0.00872665 \approx 2094.396 \text{ miles} \]

Step 3: Determine if the Beam Hits the Moon

At a divergence of approximately 2094.396 miles, compare this to the moon's radius (1000 miles). The beam diverges more than twice the radius of the moon. Since the beam deviates 2094.396 miles from the center of the moon, which is more than double its radius (1000 miles), it will not hit the moon.

Key Concepts

Radians

Radians are a way to measure angles. Unlike degrees, which divide a circle into 360 parts, radians work in terms of the circle's radius.
This means that there are exactly 2π radians in a full circle. Converting degrees to radians involves multiplying by π and dividing by 180.
This is because 180 degrees equals π radians. So, as shown in the original exercise, to convert 0.5 degrees into radians, you calculate:

• Radians = Degrees × (π/180)
• 0.5° × (π/180) ≈ 0.00872665 radians

Using radians is particularly useful in trigonometry when calculating arc lengths or when performing advanced mathematical operations since it simplifies derivatives and integrals.

Angle of Deviation

The angle of deviation is how far a path strays from its expected direction. In this exercise, the laser was meant to point directly at the moon, but due to a slight misalignment, it deviates by 0.5° from this path.
This deviation can have significant effects, especially over large distances such as the distance to the moon.
In physics and engineering, even a small angle of deviation can lead to very different outcomes than expected outcomes. Accurate measurements and alignment are crucial to avoid larger errors over distance.

Small Angle Approximation

The small angle approximation is a useful mathematical technique for simplifying problems that involve very small angles. When the angle is small (measured in radians), we can approximate

• sin(θ) ≈ θ
• tan(θ) ≈ θ

This is because, for small angles, the sine and tangent of the angle are almost equal to the angle itself (when in radians).
In this problem, the divergence distance formula is simplified by this approximation. Instead of using the entire trigonometric function, you can just multiply the angle in radians by the distance to get the divergence distance.
This saves time and effort and is accurate enough for small angles.

Divergence Distance

The divergence distance tells you how far off-target a beam or line is over a distance due to an angle of deviation.
To calculate this, as shown in the solution, you use the formula:

• D = d × θ

where D is the divergence distance, d is the total distance (in this case, to the moon), and θ is the angle of deviation in radians.
With a direct relationship, if the angle increases or the distance increases, so too does the divergence.
In practical terms, it means that a small deviation over an immense distance like earth to the moon can result in a significant misalignment, which is why precise adjustments are needed.