After such accurate advice was given by S.P.A.R.T.A.N. task force regarding the safety of the Lakeview scientists electrical equipment, you have been hired to assist the Stormchaser scientists with a top secret project. This time, in an effort to prevent future storms, you are tasked with overseeing the design and launch of an experimental government device know as a T.N.D. (Thundercloud Neutralizing Drone). This device works by remaining stationary in the center of a storm cloud, where it can be remotely triggered to neutralize the charges in the cloud and stop the storm. It is spherical in shape with a radius of $r_{TND} = 2$ m, and has a mass of $m_{TND} = 500$ kg. It also has a small propulsion system that allows it to maintain its altitude, which can be turned on remotely. In order to prepare the T.N.D. for a future storm, it will be launched upward via a massive spring to the proper height in the atmosphere where it will await a thundercloud. Once at thundercloud height, its propulsion system will hold it in place while it neutralizes the cloud.

You are tasked with carrying out the crucial step BEFORE the launch can happen: compressing the locking spring that will be used in the launch. The mechanism for how the spring is compressed is depicted in the output below:

The right (blue) sphere represents the positively charged T.N.D., which is placed on a frictionless track. The left (red) sphere represents a negatively charged T.P.D. (T.N.D. Preparation Device), which has the same mass and radius as the T.N.D., and has a net charge of $Q_{TPD} = -0.025 C$. It is held in a fixed position; its sole purpose is to attract the T.N.D. along the track towards the spring. The T.N.D. is initially held in place by the green wall, which is removable. The distance between the two gray walls is $d = 85$ m. The massive spring has a spring constant of $k_s = 4.6 \times 10^4$ N/m, and a rest length of $L_0 = 13$ m.

To complicate things, the scientists inform you of two restrictions to the safety and capabilities of the mechanism:

1. The track is not built to contain the T.N.D. at any speed; the maximum speed it should reach before hitting the spring is $v_{max} = 50$ m/s.

2. The spring is very large. In order for the compressed spring-T.N.D. system to fit on a transport truck, the spring must be compressed at least $\Delta x_{min} = 6$ m in length.

You must determine a range of charge that would be acceptable to place on the T.N.D., such that the compression process is both safe AND effective. Once compressed, the spring-T.N.D. system can be transported by truck to any desired launch location, where the T.N.D. can be remotely launched by the spring into the atmosphere to hopefully mitigate some of the crazy weather in Lakeview.

Conceptual Questions

1. Qualitatively, how does the energy change in this system? What types of energy do you have? Are they increasing or decreasing? Is energy conserved? (It may be helpful to draw energy bar charts here.)
2. What is the sign of electric potential energy in this case? What does it mean for electric potential energy to be positive or negative?
3. When the spring is fully compressed, what is the electric force on the positive charge? What is the spring force on the positive charge? What is the net force on the positive charge?
4. What is the general approach to solving a physics problem using energy? (What steps did you follow in this problem?)
5. What are the different types of energy you have worked with, and how are they quantified?
6. If a third positive charge was added to the right of the T.N.D., how would this impact your solution? What would you change? How would you calculate the new quantities?
7. What assumptions did you need to make to simplify this problem?
8. Is this problem realistic? Why or why not?