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 answer problem U, we will need to use the concepts of gravitational potential energy and kinetic energy.
a) To calculate the energy needed to lift the satellite from the Earth's surface up to a certain radius, we need to consider the change in gravitational potential energy. The formula for gravitational potential energy is given by:
PE = mgh
Where:
PE is the gravitational potential energy
m is the mass of the satellite
g is the acceleration due to gravity
h is the change in height
In this case, we are lifting the satellite from the Earth's surface to a radius of 42,300 km. This means the change in height is the difference between the radius of the orbit and the radius of the Earth. Let's calculate it step by step:
Step 1: Convert the radius of the orbit and the Earth's radius to meters:
Radius of the orbit = 42,300 km = 42,300,000 meters
Earth's radius = 6,370 km = 6,370,000 meters
Step 2: Calculate the change in height:
Change in height = Radius of the orbit - Earth's radius
Change in height = 42,300,000 meters - 6,370,000 meters
Change in height = 35,930,000 meters
Step 3: Plug the values into the formula and calculate the gravitational potential energy:
PE = mgh
PE = 50 kg * 9.8 m/s^2 * 35,930,000 meters
PE = 17,641,000,000 joules
Therefore, the energy needed to lift the satellite from the Earth's surface up to a radius of 42,300 km is 17,641,000,000 joules.
b) To calculate the kinetic energy of the satellite as it orbits, we can use the formula for kinetic energy:
KE = (1/2)mv^2
Where:
KE is the kinetic energy
m is the mass of the satellite
v is the velocity of the satellite
Step 1: Plug in the values into the formula and calculate the kinetic energy:
KE = (1/2) * 50 kg * (3076 m/s)^2
KE = 23,811,400,000 joules
Therefore, the satellite has approximately 23,811,400,000 joules of kinetic energy as it orbits.