The Moon orbits the Earth in an approximately circular path. The position of the Moon as a function of time is given by x(t) = r cos(wt) y(t) = r sin(wt), where r = 3.84 × 10° m and w = 2.46 x 10-° radians/s. What is the average velocity of the Moon measured over the interval from t = 0 to t = 6.82 days? Find its magnitude, in m/s, and find its direction, given as an angle measured counterclockwise from the positive x- axis. magnitude direction m/s ° counterclockwise from the +x-axis
The Moon orbits the Earth in an approximately circular path. The position of the Moon as a function of time is given by x(t) = r cos(wt) y(t) = r sin(wt), where r = 3.84 × 10° m and w = 2.46 x 10-° radians/s. What is the average velocity of the Moon measured over the interval from t = 0 to t = 6.82 days? Find its magnitude, in m/s, and find its direction, given as an angle measured counterclockwise from the positive x- axis. magnitude direction m/s ° counterclockwise from the +x-axis
College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
Section: Chapter Questions
Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
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Seems to have sins/cos errors that throw all the calculations. Please help for both exercises
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![**The Moon's Orbit: Calculating Average Velocity**
The Moon orbits the Earth in an approximately circular path. The position of the Moon as a function of time is given by:
\[ x(t) = r \cos(\omega t) \]
\[ y(t) = r \sin(\omega t) \]
where:
- \( r = 3.84 \times 10^8 \) m (the radius of the Moon's orbit)
- \( \omega = 2.46 \times 10^{-6} \) radians/s (the angular velocity of the Moon)
**Problem Statement:**
What is the average velocity of the Moon measured over the interval from \( t = 0 \) to \( t = 6.82 \) days? Find its magnitude, in m/s, and find its direction, given as an angle measured counterclockwise from the positive x-axis.
**Solution Steps:**
1. **Calculate the Total Time Interval:**
Convert the given time interval from days to seconds:
\[ 6.82 \text{ days} \times 24 \text{ hours/day} \times 3600 \text{ seconds/hour} \]
2. **Determine Initial and Final Positions:**
Substitute \( t = 0 \) and \( t = 6.82 \text{ days} \) into the position equations \( x(t) \) and \( y(t) \) to find initial and final coordinates.
3. **Compute Displacement:**
Calculate the difference between final and initial positions in both x and y directions to find the displacement vector.
4. **Calculate Average Velocity:**
Use the displacement vector and total time interval to compute the magnitude and direction of the average velocity.
**Required Inputs:**
- **Magnitude:** [input field] m/s
- **Direction:** [input field] ° counterclockwise from the +x-axis
By following these steps, we can derive the required values and understand the motion of the Moon in its orbit around the Earth.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F862d67f6-41ff-41a6-a751-26ec3266bc03%2F19b85baf-e2ea-4729-bfc6-d1a46bf2a7bc%2Fn0hasty.png&w=3840&q=75)
Transcribed Image Text:**The Moon's Orbit: Calculating Average Velocity**
The Moon orbits the Earth in an approximately circular path. The position of the Moon as a function of time is given by:
\[ x(t) = r \cos(\omega t) \]
\[ y(t) = r \sin(\omega t) \]
where:
- \( r = 3.84 \times 10^8 \) m (the radius of the Moon's orbit)
- \( \omega = 2.46 \times 10^{-6} \) radians/s (the angular velocity of the Moon)
**Problem Statement:**
What is the average velocity of the Moon measured over the interval from \( t = 0 \) to \( t = 6.82 \) days? Find its magnitude, in m/s, and find its direction, given as an angle measured counterclockwise from the positive x-axis.
**Solution Steps:**
1. **Calculate the Total Time Interval:**
Convert the given time interval from days to seconds:
\[ 6.82 \text{ days} \times 24 \text{ hours/day} \times 3600 \text{ seconds/hour} \]
2. **Determine Initial and Final Positions:**
Substitute \( t = 0 \) and \( t = 6.82 \text{ days} \) into the position equations \( x(t) \) and \( y(t) \) to find initial and final coordinates.
3. **Compute Displacement:**
Calculate the difference between final and initial positions in both x and y directions to find the displacement vector.
4. **Calculate Average Velocity:**
Use the displacement vector and total time interval to compute the magnitude and direction of the average velocity.
**Required Inputs:**
- **Magnitude:** [input field] m/s
- **Direction:** [input field] ° counterclockwise from the +x-axis
By following these steps, we can derive the required values and understand the motion of the Moon in its orbit around the Earth.
![### Orbital Mechanics Problem
#### Problem Statement:
A space probe approaches a planet and goes into a low orbit. If the orbiting probe's velocity is:
\[
v_x = -v \sin(\omega t),
\]
\[
v_y = v \cos(\omega t),
\]
where:
\[
v = 7.55 \times 10^3 \, \text{m/s},
\]
\[
\omega = 1.16 \times 10^{-3} \, \text{radians/s},
\]
What is the average magnitude of acceleration of the probe, in \(\text{m/s}^2\), if it remains in orbit for \(30\) minutes?
#### Answer Input:
\[
\boxed{\phantom{x}} \, \text{m/s}^2
\]
#### Explanation:
This problem involves the calculation of the average magnitude of acceleration of a space probe in a circular orbit around a planet. The given information includes the components of the velocity in the \(x\) and \(y\) directions as functions of time, the magnitude of the velocity, and the angular velocity of the probe.
Please derive the expression for the acceleration using the given velocity components and then determine the average acceleration over the given time period.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F862d67f6-41ff-41a6-a751-26ec3266bc03%2F19b85baf-e2ea-4729-bfc6-d1a46bf2a7bc%2Fprq9o1.png&w=3840&q=75)
Transcribed Image Text:### Orbital Mechanics Problem
#### Problem Statement:
A space probe approaches a planet and goes into a low orbit. If the orbiting probe's velocity is:
\[
v_x = -v \sin(\omega t),
\]
\[
v_y = v \cos(\omega t),
\]
where:
\[
v = 7.55 \times 10^3 \, \text{m/s},
\]
\[
\omega = 1.16 \times 10^{-3} \, \text{radians/s},
\]
What is the average magnitude of acceleration of the probe, in \(\text{m/s}^2\), if it remains in orbit for \(30\) minutes?
#### Answer Input:
\[
\boxed{\phantom{x}} \, \text{m/s}^2
\]
#### Explanation:
This problem involves the calculation of the average magnitude of acceleration of a space probe in a circular orbit around a planet. The given information includes the components of the velocity in the \(x\) and \(y\) directions as functions of time, the magnitude of the velocity, and the angular velocity of the probe.
Please derive the expression for the acceleration using the given velocity components and then determine the average acceleration over the given time period.
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