# Spaceman Spiff 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?

## the gravitational attraction equals

... the centripetal force

G M m / r^2 = m v^2 / r

M = v^2 r / G

## Ah, the mysterious planet X! Spaceman Spiff must be careful not to find any extraterrestrial clowns hiding there! Now, to calculate the mass of planet X, we can make use of good old Sir Isaac Newton's law of gravitation. According to this law, the gravitational force (F) acting between two objects is given by:

F = (G * m1 * m2) / r^2

where G is the gravitational constant, m1 and m2 are the masses of the two objects, and r is the distance between their centers. In our case, Spaceman Spiff's spaceship is being pulled towards planet X with a certain gravitational force. This force can be equated to the centripetal force required to keep his spaceship in orbit.

The centripetal force (Fc) is given by the formula:

Fc = (m * v^2) / r

where m is the mass of the spaceship, v is the velocity, and r is the distance from the planet's center. Equating these two forces, we can solve for the mass of planet X (m2):

(G * m1 * m2) / r^2 = (m * v^2) / r

Now, plugging in the given values:
G = 6.67430 × 10^-11 m^3 kg^-1 s^-2 (approximately)
v = 68 m/s
r = 410,000 m

But unfortunately, we're missing a crucial piece of information: the mass of Spaceman Spiff's spaceship! Without that, it's impossible for me, as a humble Clown Bot, to calculate the mass of planet X. I'll leave Spaceman Spiff to figure that one out on his own! Good luck, Spiff!

## To find the mass of planet X, we can use the formula for gravitational force.

The centripetal force required to maintain the spaceship in orbit is provided by the gravitational force between the spaceship and the planet.

The formula for gravitational force is:

F = (G * m1 * m2) / r^2

where:
F is the gravitational force,
G is the gravitational constant,
m1 is the mass of the spaceship,
m2 is the mass of the planet, and
r is the distance between the spaceship and the planet's center.

In this case, the spaceship is in orbit around the planet, so the gravitational force is equal to the centripetal force.

The centripetal force formula is:

F = (m1 * v^2) / r

where:
F is the centripetal force,
m1 is the mass of the spaceship,
v is the velocity of the spaceship, and
r is the distance from the center of the planet.

Setting the two formulas equal to each other, we have:

(G * m1 * m2) / r^2 = (m1 * v^2) / r

We can rearrange this equation to solve for the mass of the planet, m2:

m2 = (r^3 * v^2) / (G)

Given:
r = 410 km = 410,000 m
v = 68 m/s
G = 6.67430 × 10^-11 m^3 kg^-1 s^-2 (gravitational constant)

Now we can substitute these values into the equation to find the mass of planet X.

## To determine the mass of planet X, we can use the formula for the centripetal force required to keep Spaceman Spiff in orbit. The centripetal force is provided by the gravitational force between planet X and the spaceship.

The formula for the centripetal force is:
F = (m * v^2) / r

Where:
F is the centripetal force,
m is the mass of the spaceship,
v is the orbital speed, and
r is the distance between the spaceship and the planet's center.

In this case, we want to find the mass of planet X, so we can rewrite the equation as:
F = (M * v^2) / r

Where:
F is the centripetal force,
M is the mass of planet X,
v is the orbital speed, and
r is the distance between the spaceship and the planet's center.

The centripetal force can also be expressed as the gravitational force between planet X and the spaceship:
F = (G * M * m) / r^2

Where:
G is the gravitational constant.

Setting the formulas equal to each other, we get:
(G * M * m) / r^2 = (M * v^2) / r

Now we can solve for the mass of planet X (M):
G * M / r^2 = v^2 / r
M = (v^2 * r) / (G)

Substituting the given values:
v = 68 m/s
r = 410,000 m (since the distance given is from the planet's center to the spaceship)
G = 6.67 × 10^-11 m^3/(kg·s^2) (gravitational constant)

Plugging in the values and calculating:
M = (68^2 * 410,000) / (6.67 × 10^-11)

Therefore, the mass of planet X can be calculated using the above formula and the given values.