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To determine the mass of the planet, we can use the formula for gravitational force providing the centripetal force for an object in circular orbit:

\[
v^2 = \frac{G M}{r}
\]

where:
- \( v \) is the orbital speed of the object,
- \( G \) is the gravitational constant (\( 6.674 \times 10^{-11} \, \text{Nm}^2/\text{kg}^2 \)),
- \( M \) is the mass of the planet,
- \( r \) is the radius of the orbit.

Rearranging this formula to solve for \( M \):

\[
M = \frac{v^2 \cdot r}{G}
\]

### Given values:
- \( v = 4200 \, \text{m/s} \)
- \( r = 4.91 \times 10^7 \, \text{m} \)
- \( G = 6.674 \times 10^{-11} \, \text{Nm}^2/\text{kg}^2 \)

### Calculations:
1. Calculate \( v^2 \):
   \[
   v^2 = 4200^2 = 17640000 \, \text{m}^2/\text{s}^2
   \]

2. Substitute values into the equation for \( M \):
   \[
   M = \frac{17640000 \times 4.91 \times 10^7}{6.674 \times 10^{-11}}
   \]

3. Simplify the calculation.

Let's work through this.The mass of the planet is approximately \( 1.30 \times 10^{25} \) kg.

Would you like further details or have any questions?

### Related Questions:
1. How would the mass calculation change if the orbit radius increased?
2. What is the effect of a higher orbital velocity on the required mass?
3. How does gravitational force impact objects in orbit?
4. What would happen to the ship’s orbit if it slowed down?
5. Can this method be used to calculate the mass of moons or satellites?

**Tip:** Remember that the gravitational constant \( G \) is essential for calculations involving mass and gravity.

The Enterprise goes into orbit around a mysterious planet. The ship moves at 4200 m/s in a circle of radius 4.91 x 10^7 m. What is the mass of the planet?

Find out how to calculate the mass of a planet by using the orbital speed and radius of a ship in orbit. This example involves gravitational force and centripetal force, making it ideal for high school physics students.

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