I am currently working on a few problems but as of now I am stuck and unsure what to do. My confusion isn't in the mathematical computations but in the question itself. My question that I am working on is comparing the work done by propelling a single object into orbit of all 8 planets. Obviously I can look up the orbiting altitude for earth but other planets start to confuse me. I decided to use Newton's law of Universal Gravitation to find the variable d (distance between the planet and the object) but I always get a huge unreasonable number. Is there another way I can estimate the work done by launching one object into orbit around the planets? As of now I would find a function for the change in work and integrate from the planets radius to the distance I got from Newton's law of Universal Gravitation. Please help me someone.
Okay so for clarification here is one of my answers that I arrived to. This is for Venus.
$$F(x)=\frac C {x^2}$$ where C is the Constant of proportionality.
then when using the weight of the module being launched 380552.711 and the radius of Venus 6,051,000 m I get
$$380552.711=\frac C {6051000^2}$$
and $$C= 13,933,785,672,733,311,000$$
now we have the
$$\Delta W= \frac{13,933,785,672,733,311,000}{x^2}\Delta x$$
Now is the time that I would solve for the distance between the satellite and the planet. I solve for d from Newton's law of Universal Gravitation (which I suspect is the wrong thing to do and I get the distance to be $1.89218 \times 10^7$.
I use the radius of Venus as the lower bound of integration and the distance I just solved for plus the radius of venus for my upper bound. Once I solve for work by integrating the change in work function I got I get the work done to be $1.711\times 10^{13}$ joules.
Best Answer
There are some apparently open questions in regard to this problem, and I'm not sure I understand exactly what you are doing above. So I will make a suggestion for a way to approach it. Fair warning - I am going to put my satellites into orbit with zero kinetic energy - in other words, I'm only going to oppose gravity and not set them orbiting around the planets. I choose to do this because your original answer suggested you are mostly interested in that part of the problem. Using the work-energy theorem and uniform circular motion, you could add the circular motion of the satellite.
Since we can create stable orbits "anywhere" above a planet, let's assume a constant orbital distance relative to the radius of the body. If the body has a radius of $R$, let's assume each satellite orbits at a radius of $r_s=\gamma R$, where $\gamma$ is some constant. The ISS orbits at like 300 km, so $\gamma \sim (6300+300)/6300 \sim 1.05$ in that case.
Now the work required to launch a satellite into that orbit comes from the force of thrust, which is needed to oppose the force of gravity: $$F_t(p_i)=\frac{GM(p_i)m}{r^2}$$ Here I am writing "$p_i$" as "planet $i$", so $M(p_i)$ is the mass of planet $i$, I hope that is not too confusing. $m$ is the mass of our satellite and $G$ is the gravitational constant. The work required is the integral of the force from the radius of the planet to the radius of the orbit: $$W(p_i)=\int_{R(p_i)}^{r_s(p_i)} F_t(p_i) dr=GM(p_i)m\int_{R(p_i)}^{\gamma R(p_i)}\frac{dr}{r^2}$$ Notice that this illustrates one of the problems with your solution - since the force depends on the radius, you cannot simply write $\Delta W=F\Delta x$, you have to take the integral to get the right answer.
I hope this gives you enough to get started - if you still have problems I will add more details. The answer will depend on the value of $\gamma$ that you choose. This was not the only possible way to parametrize the problem. I think another reasonable way would have been to shoot for a constant orbital velocity - but you might run into some problems with flying inside a planet if you choose poorly!
Additionally, don't be worried about large numbers - the energy of a baseball being thrown by a professional pitcher is like 100 J, so launching a spaceship should easily be more than a million times that, right? Putting a baseball ($m$=150 g) into orbit (as above) takes around $10^8$ J. Expect big numbers!