A satellite is given a boost of 540 MJ of energy to move it from its initial orbit at an altitude of 150 km to a higher altitude orbit. If the satellite has a mass of 1.16 ✕ 103 kg, what is the new altitude it reaches? Take the mass of the Earth to be 5.97 x 10^24 kg and its radius to be 6.371 x 10^6 m.

To determine the new altitude the satellite reaches after receiving a boost of energy, we need to consider the conservation of mechanical energy. The mechanical energy of an object in orbit consists of kinetic energy (due to its motion) and gravitational potential energy (due to its altitude).

First, let's calculate the initial kinetic energy of the satellite. The formula for kinetic energy is:

Kinetic Energy = (1/2) * mass * velocity^2

In this case, we don't know the satellite's velocity, but we can find it using the equation for the gravitational force:

Force = (G * mass_of_Earth * mass_of_satellite) / distance^2

Where G is the gravitational constant (6.674 x 10^-11 N * m^2/kg^2), and distance is the sum of the Earth's radius and the initial altitude of the satellite.

Setting the gravitational force equal to the centripetal force (m * v^2 / distance), we can derive the equation for velocity:

v = sqrt((G * mass_of_Earth) / distance)

Plugging in the given values:

distance = Earth's radius + initial altitude = (6.371 x 10^6 m) + (150 km) = 6.371 x 10^6 m + 1.5 x 10^5 m = 6.521 x 10^6 m

mass_of_Earth = 5.97 x 10^24 kg

v = sqrt((6.674 x 10^-11 N * m^2/kg^2 * 5.97 x 10^24 kg) / (6.521 x 10^6 m)) = 7.665 x 10^3 m/s (approximately)

Now, we can calculate the initial kinetic energy using:

Kinetic Energy = (1/2) * mass * velocity^2

Kinetic Energy = (1/2) * (1.16 x 10^3 kg) * (7.665 x 10^3 m/s)^2

Kinetic Energy = 4.904 x 10^10 J

Since the total energy given to the satellite is 540 MJ = 540 x 10^6 J, we can subtract the initial kinetic energy from the total energy boost:

Energy Boost = Total Energy - Initial Kinetic Energy

Energy Boost = (540 x 10^6 J) - (4.904 x 10^10 J)

Energy Boost = 4.4596 x 10^10 J

This energy boost is equal to the increase in potential energy, so we can set it equal to mgh, where h is the change in altitude.

Energy Boost = m * g * h

h = Energy Boost / (m * g)

Plugging in the values:

h = (4.4596 x 10^10 J) / (1.16 x 10^3 kg * 9.8 m/s^2)

h = 3.851 x 10^6 m

Therefore, the new altitude the satellite reaches is 3.851 x 10^6 meters above its initial altitude of 150 km.

Keep in mind that this answer assumes that there are no other significant forces acting on the satellite during the energy boost.