Spaceman Fred orbits planet X with his spaceship. To remain in orbit at 410km from the planet's center, he should maintain a speed of 68 m/s. What is the mass of planet X?
To find the mass of planet X, we can use the formula for centripetal force and the gravitational force equation.
We know that the centripetal force required to keep Spaceman Fred in orbit is provided by the gravitational force between the planet and his spaceship. This can be expressed as:
F_centripetal = F_gravitational
The centripetal force can be calculated using the formula:
F_centripetal = m * (v^2 / r)
Where:
- F_centripetal is the centripetal force
- m is the mass of Spaceman Fred's spaceship
- v is the orbital speed of Spaceman Fred's spaceship
- r is the distance between Spaceman Fred's spaceship and the planet's center
The gravitational force can be calculated using the formula:
F_gravitational = G * (m * M) / r^2
Where:
- F_gravitational is the gravitational force
- G is the gravitational constant (approximately 6.67430 × 10^-11 m^3 kg^-1 s^-2)
- m is the mass of Spaceman Fred's spaceship
- M is the mass of planet X
- r is the distance between Spaceman Fred's spaceship and the planet's center
Setting the equations equal to each other:
m * (v^2 / r) = G * (m * M) / r^2
Now we can solve for the mass of planet X (M):
M = (r^3 * v^2) / (G * m)
Substituting the given values:
- r = 410 km = 410,000 m
- v = 68 m/s
We need to know the mass of Spaceman Fred's spaceship (m) to calculate the mass of planet X. Is the mass of Spaceman Fred's spaceship provided?