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 a satellite to a specific height using Newton's Law of Gravitation, we can follow these steps:
Step 1: Determine the initial and final positions of the satellite
Given that the satellite is being launched vertically to an orbit 800 km (or 800,000 meters) high, the initial position is at the surface of the Earth, and the final position is at an altitude of 800 km above the Earth's surface.
Step 2: Calculate the initial and final distances from the center of the Earth
The initial distance from the center of the Earth is equal to the radius of the Earth, which is 6.37 * 10^6 meters. The final distance is the sum of the radius of the Earth and the altitude of the satellite, which is (6.37 * 10^6 + 8 * 10^5) meters.
Step 3: Calculate the force of gravity at each position
To calculate the force of gravity, we can use Newton's Law of Gravitation:
F = (G * m1 * m2) / r^2
where F is the force of gravity, G is the gravitational constant, m1 and m2 are the masses of the two objects (in this case, the satellite and the Earth), and r is the distance between the centers of the two objects.
At the surface of the Earth (initial position), m1 is the mass of the satellite (1500 kg), m2 is the mass of the Earth (5.98 * 10^24 kg), and r is the radius of the Earth (6.37 * 10^6 meters).
Plugging the values into the equation gives us:
F_initial = (6.67 * 10^-11 Nm^2/kg^2 * 1500 kg * 5.98 * 10^24 kg) / (6.37 * 10^6 meters)^2
At the final position, m1 is still the mass of the satellite (1500 kg), m2 is still the mass of the Earth (5.98 * 10^24 kg), and r is the sum of the radius of the Earth and the altitude of the satellite (6.37 * 10^6 + 8 * 10^5) meters.
Plugging the values into the equation gives us:
F_final = (6.67 * 10^-11 Nm^2/kg^2 * 1500 kg * 5.98 * 10^24 kg) / (6.37 * 10^6 meters + 8 * 10^5 meters)^2
Step 4: Calculate the work required
To calculate the work required, we use the equation:
W = ∫F * dr
where W is the work and dr is the change in position. Since the satellite is launched vertically, the force and the change in position are in the same direction, so we can simplify the integration to:
W = ∫F * dr = ∫F * d(r_initial - r_final) = ∫(F_final - F_initial) * dr
Now we need to evaluate the definite integral of (F_final - F_initial) with respect to r from r_initial to r_final. Integrating (F_final - F_initial) gives us:
W = (F_final - F_initial) * (r_final - r_initial)
Plugging in the values we calculated earlier, we have:
W = (F_final - F_initial) * (r_final - r_initial)
= (F_final - F_initial) * ((6.37 * 10^6 + 8 * 10^5) - 6.37 * 10^6)
Now we can substitute the values we calculated for F_initial, F_final, and the distances into the equation to find the work required.
Work= INT Fdx
= INT GMeMs/x^2 dx
integrate from re to re+800,000m
I will be happy to check your work.