Hohman orbits University Physics I

[THTremere]

Homework Statement

Im not sure how to solve this one...

Hohmann orbits are the lowest energy orbits to move things from one planet to another in a solar system. In order to send something from the Earth to another planet in the solar system, you launch the object when the Earth and the other planet are laid out at the opposite ends of the major axis of an ellipse which has the sun at one focus. Let's see how useful Hohmann orbits can be launching something from the Earth to Mars. Take th distance from the Earth to the sun to be 1.5x10^11m, and from Mars to the sun to be 2.3x10^11m. The mass of the sun is 2x10^30kg.
a) Let's say that the total energy required for the Hohmann orbit around the sun from the Earth to Mars for a probe is -3x10^11 J. What is the mass of a probe that can achieve this energy requirement?


I am thinking it involves setting up E tot= K + U(gravitational) But I am not sure really where to go from there! I don't have a velocity...maybe use conservation of angular momentum...Anyway I am not sure how to even set up this problem it seems. Any help in the right direction would be greatly appreciated.

Homework Equations

E=K+U
E=1/2mV^2 - GMm/r
L=L (angular momentum)

The Attempt at a Solution

Im not sure how to set this one up...If I could get a shove in the right direction I could really attempt this :)

I tried this ... mp(mass of the probe)=E-total/(1/2V^2-GM(earth)/r(earth)-GM(sun)/R(sun)-GM(mars)/r(mars)
I doubt that is correct though...

 

[anathema]

Hello, it seems like you are on the right track with using the equation E=K+U. To find the mass of the probe, you will need to rearrange the equation to solve for mass. This can be done by setting the total energy requirement equal to the kinetic energy (since the probe will have no potential energy at the beginning and end of its journey).

So the equation would be E tot = 1/2mv^2. You can then plug in the given values for the energy requirement and the distance between the Earth and Mars to solve for the velocity of the probe. From there, you can use the equation for kinetic energy to solve for the mass of the probe.

As for using conservation of angular momentum, that would not be necessary in this problem since the Hohmann orbit is a two-body problem (only considering the Earth and Mars). Conservation of angular momentum would come into play if there were more than two bodies involved, such as if you were trying to send the probe to a planet further out in the solar system. Hope this helps and good luck!

 

[nlightner]

As a scientist, it is important to approach a problem in a systematic and logical manner. In this case, we are given the energy requirement for a Hohmann orbit from Earth to Mars and we need to determine the mass of a probe that can achieve this energy requirement.

First, we need to understand the concept of Hohmann orbits and how they work. As mentioned in the problem statement, Hohmann orbits are the lowest energy orbits to move objects from one planet to another in a solar system. In order to achieve this, the object is launched from Earth when it and Mars are at opposite ends of an ellipse around the sun, with the sun at one focus.

Now, let's look at the equations we have been given. The total energy required for a Hohmann orbit is given by the equation E_tot=K+U, where E_tot is the total energy, K is the kinetic energy, and U is the gravitational potential energy. We also have the equation for gravitational potential energy, U=-GMm/r, where G is the gravitational constant, M is the mass of the larger body (in this case, the sun), m is the mass of the smaller body (probe), and r is the distance between the two bodies.

From these equations, we can see that the energy required for a Hohmann orbit depends on the masses of the two bodies, the distance between them, and the velocity of the smaller body. However, we are not given the velocity in this problem. So, how can we solve for the mass of the probe?

One approach is to use the concept of conservation of energy. In a Hohmann orbit, the total energy is constant, meaning it remains the same throughout the entire orbit. This means that the energy required for a Hohmann orbit from Earth to Mars is equal to the energy required for a Hohmann orbit from Mars to Earth. So, we can set up an equation where the energy required for the Hohmann orbit from Earth to Mars is equal to the energy required for the Hohmann orbit from Mars to Earth. This equation would look like this:

E_tot (Earth to Mars) = E_tot (Mars to Earth)

Substituting in the values given in the problem statement, we get:

-3x10^11 J = 1/2mV^2 - GMm/r (Earth to Mars) = 1/2mV^2 - GMm