The position of the Moon as a function of time is given by the parametric equations:

$x(t) = r \cos(\omega t)$ $y(t) = r \sin(\omega t)$

Where:

• $r = 3.84 \times 10^8 \, \text{m}$
• $\omega = 2.46 \times 10^{-6} \, \text{rad/s}$

Step 1: Convert the time interval to seconds

The given time interval is $t = 6.9 \, \text{days}$. We first need to convert this into seconds: $t = 6.9 \, \text{days} \times 24 \, \text{hours/day} \times 3600 \, \text{seconds/hour}$

Step 2: Determine the positions at $t = 0$ and $t = 6.9$ days

At $t = 0$, the position of the Moon is: $x(0) = r \cos(0) = r = 3.84 \times 10^8 \, \text{m}$ $y(0) = r \sin(0) = 0$

At $t = 6.9 \, \text{days}$, the position of the Moon is: $x(6.9) = r \cos(\omega t)$ $y(6.9) = r \sin(\omega t)$ Substitute $\omega t = 2.46 \times 10^{-6} \times t$.

Step 3: Calculate the displacement vector

The displacement vector over the time interval is: $\Delta x = x(6.9) - x(0)$ $\Delta y = y(6.9) - y(0)$

Step 4: Compute the average velocity

The average velocity magnitude is given by: $v_{\text{avg}} = \frac{\sqrt{(\Delta x)^2 + (\Delta y)^2}}{\Delta t}$

Where $\Delta t = 6.9 \, \text{days}$ converted to seconds.

Step 5: Direction of the average velocity

The direction is the angle $\theta$ of the displacement vector relative to the positive x-axis: $\theta = \tan^{-1}\left(\frac{\Delta y}{\Delta x}\right)$

Let me calculate these values for you.It looks like I made an error in applying trigonometric functions. I will correct this by using the correct syntax for cosine and sine. Let me recalculate the values.It seems like I can’t do more advanced data analysis right now. Please try again later.

If you need further assistance, feel free to ask. Let me know if you would like to go over the problem conceptually or if you want me to guide you through the manual calculations.

Here are 5 relative questions to expand this concept:

1. How does the angular velocity $\omega$ affect the Moon's orbital period?
2. What is the formula for instantaneous velocity in circular motion?
3. How can we calculate the centripetal acceleration of the Moon given the parameters?
4. What role does the gravitational force play in maintaining the Moon's orbit?
5. How can we estimate the time it takes for the Moon to complete one full orbit?

Tip: The direction of velocity in circular motion is always tangent to the orbit at any given point.