Problem 84 - Precalculus with Calculus Previews

Question

Planet Mercury The planet Mercury completes one rotation on its axis every 59 days. Through what angle (measured in degrees) does it rotate in (a) 1 day \(,\) (b) 1 hour, and (c) 1 minute?

Step-by-Step Solution

Answer

Mercury rotates approximately 6.10° in 1 day, 0.2542° in 1 hour, and 0.00424° in 1 minute.

Step 1: Understanding Full Rotation

To solve the problem, we first need to understand that Mercury rotates 360 degrees in 59 days because one complete rotation corresponds to a full circle.

Step 2: Calculate Rotation in 1 Day

To find out how much Mercury rotates in 1 day, we divide the total degrees (360°) by the number of days for a full rotation (59 days): \[ \text{Degrees per day} = \frac{360}{59} \approx 6.10 \text{ degrees} \]

Step 3: Calculate Rotation in 1 Hour

There are 24 hours in a day, so to find the rotation in 1 hour, we divide the degrees of rotation per day by 24: \[ \text{Degrees per hour} = \frac{6.10}{24} \approx 0.2542 \text{ degrees} \]

Step 4: Calculate Rotation in 1 Minute

There are 60 minutes in an hour. To find the rotation in 1 minute, we divide the degrees of rotation per hour by 60: \[ \text{Degrees per minute} = \frac{0.2542}{60} \approx 0.00424 \text{ degrees} \]

Key Concepts

Planetary Rotation

Planetary rotation refers to the spinning motion of a planet around its axis. This axis is an imaginary line that runs through the planet's poles. Every planet, including Mercury, has its own unique rotation period, which is the time it takes to complete one full rotation. This rotation is crucial for determining the length of a day on that planet. For Mercury, a full rotation takes 59 Earth days, making its days extremely long compared to Earth. The planetary rotation impacts the planet's weather patterns, solar exposure, and overall environment.

Understanding planetary rotation helps scientists learn more about a planet's natural phenomena, such as seasons and climate. Observations and calculations related to planetary rotation are important not only in astronomy but also in planning space missions and interpreting data gathered from these bodies. Continuous study of planetary rotation enriches our knowledge of the universe and its various mechanisms.

Angular Measurement

Angular measurement is an essential concept in trigonometry and astronomy. A full rotation around a planet's axis is equivalent to 360 degrees. This concept simplifies the calculation of angles that occur as a planet rotates and as celestial objects appear to move through the sky. When we talk about how much a planet like Mercury rotates in terms of degrees, we're referring to angular measurements.

To find the angle through which Mercury rotates each day, we divide the full circle (360 degrees) by the number of days it takes for a complete rotation, which is 59. This calculation shows that Mercury rotates approximately 6.10 degrees per day. Similarly, you can further divide to find rotations per hour and per minute.

• You divide the daily rotation by 24 to find degrees rotated per hour.
• You then divide the hourly rotation by 60 to determine degrees rotated per minute.

Angular measurement allows us to understand and predict various astronomical phenomena and is key in many scientific and practical applications.

Units of Time Conversion

Time conversion is necessary when measuring planetary movements to align them with our timekeeping systems. Earthly time is typically segmented into units like days, hours, and minutes for the sake of convenience and consistency in our daily lives. When studying planetary rotations, like that of Mercury, we often need to work with these units to calculate rotations over shorter time intervals.

For instance, once we know how much Mercury rotates in a day, we convert that into smaller time units to further understand its rotation.

• There are 24 hours in one day so it's divided further to find rotations per hour.
• From hours, we convert to minutes, knowing there are 60 minutes in an hour.

Understanding time conversions aids in comparative studies and ensures that calculations across different contexts are accurate and reflect the correct time spans. These conversions are critical in domains such as navigation, space exploration, and even daily timekeeping.