Launch Package from Spinning Asteroid: Calculating Required Spring Compression

[fball558]

here is the problem
A package of mass 6 kg sits at the equator of an airless asteroid of mass 5.7e5 kg and radius 41 m, which is spinning so that a point on the equator is moving with speed 4 m/s. We want to launch the package in such a way that it will never come back, and when it is very far from the asteroid it will be traveling with speed 230 m/s. We have a large and powerful spring whose stiffness is 1.4e5 N/m. How much must we compress the spring?

not really sure where to start but what i tried was finding the escape velocity of the box by using sqrt(2GM/r) where G is the gravitation attraction between the two masses. then try to find how much work would need to be done to make the box move that fast and relate that work to the spring compression. but got the wrong answer. any help would be great.

 

[fball558]

never mind found it out. just had to assume that "far away" means no patential energy. it is traveling at 230 so find only kenetic. them use potential energy equation of spring 1/2ks^2 solve for s (spring stretch) and that is your answer. ends up being something like 1.5 meter or 1.05 forget what it was.

 

[nlightner]

As a scientist, my recommendation would be to approach this problem using the principle of conservation of energy. First, we can calculate the potential energy of the package at the surface of the asteroid using the formula U = mgh, where m is the mass of the package, g is the acceleration due to gravity, and h is the height from the surface of the asteroid. This will give us the amount of energy needed to lift the package off the surface of the asteroid.

Next, we can calculate the kinetic energy of the package at the equator of the asteroid using the formula KE = 1/2mv^2, where m is the mass of the package and v is the speed of the asteroid at the equator. This will give us the amount of energy that the package already has due to the asteroid's rotation.

To launch the package with a speed of 230 m/s, we need to add the necessary amount of energy to its existing kinetic energy. This can be achieved by compressing the spring and releasing it, converting the potential energy stored in the spring into kinetic energy of the package.

Using the formula for potential energy stored in a spring, PE = 1/2kx^2, where k is the stiffness of the spring and x is the compression distance, we can calculate the necessary compression distance to achieve the desired launch velocity.

By equating the total energy needed to launch the package (potential energy at the surface + kinetic energy at the equator + energy from the compressed spring) with the final kinetic energy of the package (230 m/s), we can solve for x and determine the required compression distance.

I hope this approach helps you solve the problem. If you encounter any difficulties, I suggest breaking down the problem into smaller, more manageable steps and double-checking your calculations. Good luck!