Use Newton's Law of Gravitation to compute the work W required to launch a 1500 kg satellite vertically to an orbit 800 km high. You may assume that the earth's mass is 5.98*10^24 kg and is concentrated at its center. Take the radius of the earth to be 6.37*10^6 m and G = 6.67*10^-11 Nm^2/kg^2.
To compute the work required to launch the satellite, we need to find the change in potential energy as the satellite moves from the surface of the Earth to its final orbit.
The formula for the gravitational potential energy is given by:
PE = - G * (m1 * m2) / r
Where:
PE is the potential energy
G is the gravitational constant (6.67*10^-11 Nm^2/kg^2)
m1 and m2 are the masses of the two objects (in this case, the satellite and the Earth)
r is the distance between the centers of the two objects
In this case, the satellite's mass (m1) is 1500 kg, the mass of the Earth (m2) is 5.98*10^24 kg, and the initial distance (r1) is the radius of the Earth (6.37*10^6 m). The final distance (r2) can be calculated by adding the height of the orbit (800 km or 8 * 10^5 m) to the radius of the Earth.
So, r2 = r1 + height = 6.37*10^6 m + 8*10^5 m = 7.17*10^6 m
Substituting these values into the formula, we get:
PE1 = - G * (m1 * m2) / r1
PE2 = - G * (m1 * m2) / r2
The work done is equal to the change in potential energy:
W = PE2 - PE1
Now, we can calculate the work:
PE1 = - (6.67*10^-11 Nm^2/kg^2) * (1500 kg * 5.98*10^24 kg) / (6.37*10^6 m)
PE2 = - (6.67*10^-11 Nm^2/kg^2) * (1500 kg * 5.98*10^24 kg) / (7.17*10^6 m)
W = PE2 - PE1
Plug in these values into a calculator or use a computer program to get the numerical answer.