Math Problem Statement

The planet XYZ travels about the star ABC in an orbit that is almost circular. Assume that the orbit is a circle with radius 82,000,000 miles. Assume there are 24 hours in one day on planet XYZ. (a) Assume that XYZ planet year is 378 days, and find the angle formed by XYZ's movement in one day. (b) Give the angular speed in radians per hour. (c) Find the linear speed of XYZ in miles per hour.

Solution

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We are given the following details:

  • The planet XYZ orbits star ABC in a nearly circular orbit with a radius of 82,000,00082,000,000 miles.
  • A day has 24 hours.
  • The planet XYZ takes 378 days to complete one revolution (one full orbit) around the star.

We are tasked with finding:

(a) The angle formed by XYZ’s movement in one day

To calculate the angle formed in one day, we need to understand that XYZ completes one full revolution (360 degrees or 2π2\pi radians) in 378 days. Therefore, the angle moved in one day is:

θ=2π378 radians/day\theta = \frac{2\pi}{378} \text{ radians/day}

(b) The angular speed in radians per hour

To find the angular speed per hour, we take the angular speed per day (from part a) and divide it by 24 (since there are 24 hours in a day):

ω=θ24=2π378×24 radians/hour\omega = \frac{\theta}{24} = \frac{2\pi}{378 \times 24} \text{ radians/hour}

(c) The linear speed in miles per hour

The linear speed vv can be calculated using the formula:

v=rωv = r \cdot \omega

Where:

  • r=82,000,000r = 82,000,000 miles (the radius of the orbit),
  • ω\omega is the angular speed in radians per hour (from part b).

Let me calculate the exact values for each part now.Here are the results for each part:

(a) The angle formed by XYZ's movement in one day:

θ0.0166 radians/day\theta \approx 0.0166 \text{ radians/day}

(b) The angular speed in radians per hour:

ω0.000693 radians/hour\omega \approx 0.000693 \text{ radians/hour}

(c) The linear speed of XYZ in miles per hour:

v56,792 miles/hourv \approx 56,792 \text{ miles/hour}

Would you like further clarification on any of these steps or more details? Here are some additional related questions:

  1. How would the angular speed change if the orbit radius was doubled?
  2. How is the linear speed related to the radius of the orbit?
  3. What would happen to the angular speed if the planet completed the orbit in fewer days?
  4. How can we convert angular speed from radians per hour to degrees per hour?
  5. What would be the linear speed if the planet's orbital radius was halved?

Tip: To easily convert angular speed from radians to degrees, remember that 2π2\pi radians is equivalent to 360 degrees.

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Math Problem Analysis

Mathematical Concepts

Angular Speed
Linear Speed
Circular Motion

Formulas

θ = 2π / T
ω = θ / t
v = r * ω

Theorems

Angular velocity formula
Linear velocity formula

Suitable Grade Level

Grades 10-12