Mars' moon Phobos has a mass of 1.07x1016 kg and a mean radius
of 11.1 km. Let's say our rocket blasts off from the surface of Phobos
when Phobos is 9500 km from the surface of Mars. Does the rocket have
to work harder to escape from the gravitational pull of Phobos or that of
Mars?
The energy per unit mass required to escape the gravitational pull is
G M/R
where M is the mass of the planet or moon that is being escaped, and R is the radius (measured from the center) where the launch process starts. Compare the M/R values for Phobos and Mars. You will have to look up the mass of Mars and add the radius of Mars to 9500 km to get the R value for Mars. G is the universal gravity constant, which is the same for both.
To determine whether the rocket has to work harder to escape from the gravitational pull of Phobos or Mars, we need to compare the gravitational forces exerted by each body.
The gravitational force exerted by a celestial body can be calculated using Newton's law of gravitation:
F = (G * M1 * M2) / r^2
Where:
F is the gravitational force
G is the gravitational constant (approximately 6.67430 × 10^-11 N(m/kg)^2)
M1 and M2 are the masses of the two bodies
r is the distance between the centers of the two bodies
Let's calculate the gravitational force exerted by Phobos and Mars at the surface of Phobos:
Gravitational force exerted by Phobos on the rocket:
F_phobos = (G * M_rocket * M_phobos) / r_phobos^2
Gravitational force exerted by Mars on Phobos:
F_mars_phobos = (G * M_mars * M_phobos) / r_mars_phobos^2
Where:
M_rocket is the mass of the rocket.
M_mars = 6.39 × 10^23 kg (mass of Mars)
M_phobos = 1.07 × 10^16 kg (mass of Phobos)
r_phobos = 11.1 km
r_mars_phobos = 9500 km + 11.1 km (distance from Phobos' surface to the surface of Mars plus the radius of Phobos)
Let's calculate these values:
F_phobos = (6.67430 × 10^-11 N(m/kg)^2 * M_rocket * 1.07 × 10^16 kg) / (11.1 km)^2
F_mars_phobos = (6.67430 × 10^-11 N(m/kg)^2 * 6.39 × 10^23 kg * 1.07 × 10^16 kg) / (9500 km + 11.1 km)^2
Now, compare the magnitudes of the two forces. If F_phobos > F_mars_phobos, then the rocket has to work harder to escape from Phobos' gravitational pull. If F_phobos < F_mars_phobos, then the rocket has to work harder to escape from Mars' gravitational pull.
Note: The mass of the rocket is needed to complete the calculation, but it has not been provided in the question.
To determine whether the rocket needs to work harder to escape the gravitational pull of Phobos or Mars, we need to compare the gravitational forces exerted by both objects on the rocket.
First, let's calculate the gravitational force exerted by Phobos on the rocket:
The gravitational force between two objects can be calculated using the formula:
F = G * (m1 * m2) / r^2
Where
F is the gravitational force,
G is the gravitational constant (6.67 x 10^-11 N m^2 / kg^2),
m1 and m2 are the masses of the two objects, and
r is the distance between the centers of the two objects.
For Phobos, given:
Mass of Phobos (m1) = 1.07 x 10^16 kg
Mean radius of Phobos (r) = 11.1 km = 11,100 m
First, we need to convert the mean radius of Phobos to the distance between the rocket and Phobos by subtracting the radius of Phobos from the distance between Phobos and Mars:
Distance between rocket and Phobos = Distance between Phobos and Mars - Radius of Phobos
Given:
Distance between Phobos and Mars = 9500 km = 9,500,000 m
So,
Distance between rocket and Phobos = 9,500,000 m - 11,100 m = 9,488,900 m
Now we can calculate the gravitational force exerted by Phobos on the rocket:
F_Phobos = (6.67 x 10^-11 N m^2 / kg^2) * ((1.07x10^16 kg * m2) / (9,488,900 m)^2)
Next, let's calculate the gravitational force exerted by Mars on the rocket:
Given:
Mass of Mars = 6.42 x 10^23 kg
Radius of Mars = 3,389.5 km = 3,389,500 m
The distance between the rocket and Mars is given as 9500 km = 9,500,000 m.
Now we can calculate the gravitational force exerted by Mars on the rocket:
F_Mars = (6.67 x 10^-11 N m^2 / kg^2) * ((6.42x10^23 kg * m2) / (9,500,000 m)^2)
Comparing the values of F_Phobos and F_Mars will determine which gravitational force the rocket needs to overcome. If F_Phobos is greater than F_Mars, the rocket needs to work harder to escape Phobos' gravitational pull. Otherwise, if F_Phobos is smaller than F_Mars, the rocket needs to work harder to escape Mars' gravitational pull.