Let's go step by step to solve the given questions.

Given Data:

• Radius of orbit $r = 78,000,000$ miles.
• Length of the year on planet XYZ $T = 322$ days.
• 1 day on XYZ = 24 hours.

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

The angle formed by XYZ's movement in one day is related to the total angle around the orbit, which is $2\pi$ radians (since a full revolution is $360^\circ$ or $2\pi$ radians).

Since the planet's year consists of 322 days, the angle for one complete revolution $2\pi$ radians will be covered over 322 days. Therefore, the angle moved in one day is:

$\text{Angle per day} = \frac{2\pi}{322}$

This simplifies to:

$\text{Angle per day} = \frac{\pi}{161} \, \text{radians}$

(b) Angular speed in radians per hour

The angular speed is the angle traversed per unit of time. Here, we know the planet moves $\frac{\pi}{161}$ radians in one day. Since one day equals 24 hours, the angular speed in radians per hour is:

$\text{Angular speed} = \frac{\pi}{161 \times 24} = \frac{\pi}{3864} \, \text{radians per hour}$

(c) Linear speed in miles per hour

The linear speed can be found by using the formula:

$v = \frac{\text{circumference}}{\text{time period}}$

where the circumference $C$ of the orbit is:

$C = 2\pi r$

Substituting the radius $r = 78,000,000$ miles:

$C = 2\pi \times 78,000,000 = 156,000,000\pi \, \text{miles}$

The time period for one complete revolution is the number of hours in a year, which is:

$\text{Time period} = 322 \, \text{days} \times 24 \, \text{hours/day} = 7728 \, \text{hours}$

Now, the linear speed $v$ is:

$v = \frac{156,000,000\pi}{7728} \approx 63,567.1 \, \text{miles per hour}$

So, the linear speed is approximately 63,567 miles per hour.

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Final Answers:

• (a) The angle formed by XYZ's movement in one day is $\frac{\pi}{161}$ radians.
• (b) The angular speed is $\frac{\pi}{3864}$ radians per hour.
• (c) The linear speed is approximately 63,567 miles per hour.

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Would you like to see further details or explanations for any of these steps? Or have any additional questions?

Additional Questions to Expand:

1. How would the angular speed change if the planet's year were longer or shorter?
2. How does the linear speed depend on the radius of the orbit?
3. What happens to the angle moved per day if the year on XYZ is increased?
4. How do you calculate the period of an orbit from the linear speed?
5. How would the time taken to complete one orbit change if the radius was halved?

Tip:

To better understand angular speed, remember that it’s a measure of how much angle (in radians) an object covers per unit of time. It’s particularly useful for circular or orbital motion.