Minimum Energy Required to Launch a Satellite into Orbit

To calculate the minimum energy required to launch a satellite into a circular orbit at a height 2R from the Earth's surface, we need to consider the gravitational potential energy and the kinetic energy of the satellite.

1. Gravitational Potential Energy:
The gravitational potential energy of an object depends on its mass (m), the acceleration due to gravity (g), and the height (h) above the reference point. In this case, the reference point is the Earth's surface.

The gravitational potential energy (PE) is given by the formula:
PE = mgh

In this case, the height (h) is given as 2R, where R is the radius of the Earth. So, the gravitational potential energy can be written as:
PE = mgh = mg(2R)

2. Kinetic Energy:
The kinetic energy (KE) of an object depends on its mass (m) and its velocity (v). In this case, the satellite is in a circular orbit, so its velocity can be calculated using the formula for the orbital speed.

The orbital speed (v) is given by the formula:
v = √(GM/R)

Where G is the universal gravitational constant, M is the mass of the Earth, and R is the distance between the center of the Earth and the satellite's orbit.

The kinetic energy can be calculated using the formula:
KE = (1/2)mv²

Since the mass (m) cancels out, we can write the kinetic energy as:
KE = (1/2)mv² = (1/2)m(√(GM/R))² = (1/2)m(GM/R)

3. Total Energy:
The total energy (E) required to launch the satellite into orbit is the sum of the gravitational potential energy (PE) and the kinetic energy (KE).

E = PE + KE = mg(2R) + (1/2)m(GM/R)

Since we are calculating the minimum energy required, we need to minimize this total energy expression.

4. Solving for Minimum Energy:
To minimize the energy expression, we can take the derivative of the total energy (E) with respect to the height (h) and set it equal to zero.

dE/dh = 0

By solving this equation, we can find the minimum energy required to launch the satellite into orbit.

5. Conclusion:
In this question, we have discussed the calculation of the minimum energy required to launch a satellite into a circular orbit at a height 2R from the Earth's surface. The total energy is calculated by considering the gravitational potential energy and the kinetic energy of the satellite. By minimizing the total energy expression, we can find the minimum energy required for the launch.