Why Is Achieving Orbit Considered Halfway to Anywhere?

[dswatson]

The science fiction writer Robert Heinlein once said, "If you can get into orbit, then you're halfway to anywhere". Justify this statement by comparing the minimum energy needed to place a satellite into low Earth orbit (h=400km) to that needed to set it completely free from the bonds of Earth's gravity. Neglect any effects due to air resistance.

E = KE + U
E = 1/2mv^2 + U
U = -(int)[F*dr]
U = -G(Mm/r^2)
E = m( (1/2)v^2 - G(M/r^2)

m = mass satellite
M = mass earth

Im stuck because I am not given a mass for the satellite and I know that excape velocity does not depend on mass. It is approxmately 7mi/s but I am unsure of how to show that energy is not mass dependent and then solving for the equation with no mass.

 

[Irid]

Well, first of all, the total energy is

$$E = m\left( \frac{v^2}{2} - \frac{GM}{r}\right)$$

(you got the power wrong). It depends on m, but if you set it equal to zero E=0, then m cancels and you can solve for v without knowing m.

 

[thomate1]

I would approach this problem by first clarifying the statement and the question being asked. The statement by Robert Heinlein is referring to the fact that once an object is in orbit, it has achieved a certain amount of energy that allows it to travel to other destinations in space more easily. This is because the object is no longer fighting against Earth's gravity, but is instead using it to its advantage in a stable orbit. Therefore, the minimum energy required to place a satellite into low Earth orbit is significantly less than the energy needed to completely escape Earth's gravity.

To calculate the minimum energy needed to place a satellite into low Earth orbit, we can use the equation E = KE + U, where E is the total energy, KE is the kinetic energy, and U is the potential energy. In this case, we can neglect any effects due to air resistance, so we can focus on the potential energy component.

Using the equation U = -G(Mm/r^2), where G is the gravitational constant, M is the mass of Earth, m is the mass of the satellite, and r is the distance between the center of Earth and the satellite, we can calculate the potential energy for both scenarios.

For a satellite in low Earth orbit, r would be equal to the radius of Earth plus the altitude of the orbit (400km). This would result in a smaller value for r compared to the distance between Earth and an object completely escaping its gravity. Therefore, the potential energy for a satellite in low Earth orbit would be lower.

To compare the minimum energy needed to place a satellite into low Earth orbit to that needed to completely escape Earth's gravity, we can consider the equation E = m((1/2)v^2 - G(M/r^2)), where v is the velocity of the satellite. As mentioned, the escape velocity for Earth is approximately 7 miles per second. This velocity is not dependent on the mass of the satellite, as you correctly stated. Therefore, we can solve for the equation without considering mass.

E = ((1/2)(7 mi/s)^2 - G(M/r^2))

As we can see, the minimum energy needed to completely escape Earth's gravity is significantly higher than the energy needed to place a satellite into low Earth orbit. This justifies the statement by Robert Heinlein, as once an object is in orbit, it has already achieved a significant amount of energy that can be used to travel to other destinations in space