Consider problem U concerning the geosynchronous satellite of mass 50 kg. It orbits at a radius of 42,300 km with a speed of 3076 m/s. a) How much energy is needed to lift the satellite from the earth's surface (Re=6,370 km) up to that radius? b) How much kinetic energy does the satellite have as it orbits?Consider problem U concerning the geosynchronous satellite of mass 50 kg. It orbits at a radius of 42,300 km with a speed of 3076 m/s. a) How much energy is needed to lift the satellite from the earth's surface (Re=6,370 km) up to that radius? b) How much kinetic energy does the satellite have as it orbits?
To solve this problem, we need to understand the concepts of potential and kinetic energy.
a) To calculate the energy needed to lift the satellite from the Earth's surface to the given radius, we can use the principle of conservation of energy. The energy required is equal to the difference in potential energy of the satellite at the final and initial positions.
The potential energy of an object at a certain radius from the center of the Earth is given by the formula:
PE = -G * (m1 * m2) / r
Where G is the gravitational constant, m1 is the mass of the Earth, m2 is the mass of the satellite, and r is the radius from the Earth's center.
First, let's calculate the potential energy at the Earth's surface (initial position). The radius at the Earth's surface is Re = 6,370 km = 6,370,000 m.
PE_initial = -G * (m1 * m2) / Re
Next, let's calculate the potential energy at the given radius (final position). The radius at the given position is given as 42,300 km = 42,300,000 m.
PE_final = -G * (m1 * m2) / r
The energy needed to lift the satellite is the difference between the initial and final potential energy:
Energy = PE_final - PE_initial
b) The kinetic energy of an object in motion is given by the formula:
KE = (1/2) * m * v^2
Where m is the mass of the object, and v is its velocity.
The kinetic energy of the satellite as it orbits is given by:
KE = (1/2) * m * v^2
Where m is the mass of the satellite, and v is its orbital velocity at the given radius.
Now, let's substitute the given values into the equations and calculate the answers.
a) First, calculate the potential energy:
PE_initial = -G * (m1 * m2) / Re
PE_initial = - (6.67430 × 10^-11 m^3 kg^-1 s^-2) * ((5.972 × 10^24 kg) * (50 kg)) / (6,370,000 m)
PE_initial = -3.178299 × 10^10 J
PE_final = -G * (m1 * m2) / r
PE_final = - (6.67430 × 10^-11 m^3 kg^-1 s^-2) * ((5.972 × 10^24 kg) * (50 kg)) / (42,300,000 m)
PE_final = -1.0831406 × 10^10 J
Energy = PE_final - PE_initial
Energy = (-1.0831406 × 10^10 J) - (-3.178299 × 10^10 J)
Energy = -2.0951584 × 10^10 J
Therefore, the energy needed to lift the satellite from the Earth's surface to the given radius is approximately -2.0951584 × 10^10 J.
b) Now, calculate the kinetic energy:
KE = (1/2) * m * v^2
KE = (1/2) * (50 kg) * (3076 m/s)^2
KE = 2.3527 × 10^8 J
Therefore, the kinetic energy of the satellite as it orbits is approximately 2.3527 × 10^8 J.