Project 12B: Saving a probe

1. Consider the statement: Two objects collide and bounce off each other, therefore this MUST be an elastic collision. Is this statement true? Why or why not?
2. What is an elastic collision? What is conserved in this type of collision?
3. What is an inelastic collision? What is conserved in this type of collision?
4. What is a maximally inelastic collision? How does this differ from the more general inelastic collision?

You are trying to recover HAL. HAL, if you remember, was a part of the satellite ($m_{\rm t}=4500\,{\rm kg}$) that the Carver Media Group Network (CMGN) launched. The satellite (and HAL) was designed to communicate with Earth out to a distance of 3.8 million kilometers. As part of the electronics HAL contains a green and a red light-emitting diode (LED) mounted on the outer surface of the satellite. One of the probes ($m_{\rm p}=400\,{\rm kg}$) remains attached to a single, very stiff spring ($k_{\rm p}=5.3\times10^{9}{\rm N/m}$) that can be compressed remotely and then released to fire off the probe. You have hacked into this remote firing mechanism.

Unfortunately, a transcription error was made by Mr. Stamper, Carver's chief “engineer” when the satellite was initially launched. As a result, the satellite escaped Earth's gravity and is currently traveling in a straight line away from the Earth at a distance of 1.9 million kilometers. Its speed is nearly constant at $340\,{\rm m/s}$. The gyroscope system that keeps the orientation of the satellite constant is still working. However, an asteroid ($m_{\rm a}=9300\,{\rm kg}$) traveling at a speed of $950\,{\rm m/s}$ is on a direct collision course (in line with the Earth and HAL) and the collision is imminent. The asteroid is presently 500 kilometers from the satellite.

Your team can recapture the satellite if it can be returned to Earth. You should design a way to return the satellite to Earth. You will also need to ensure the asteroid will not collide with the satellite - damaging the probe is ok. Your team also needs to determine the minimum amount of time until the possible collision to determine if communication from Earth can be completed in time.

• What did you assume about the spring? Is this realistic? Why or why not?
• If you assume the asteroid-probe collision were elastic, how would this change your approach for this interaction?
• In this problem, you are told the satellite escaped Earth's gravity, and used this in developing your model. Is this a good approximation?
• If you include the Earth as part of your system to include its impact on your results, how fast must the asteroid be moving away from Earth to make sure it is never a danger to the people on Earth? How fast must the probe be moving in order to accomplish this?
• If we include Earth and its impact on the motion of the probe, with what velocity must it leave the transmitter so that it is moving fast enough to ensure the asteroid is never a threat to Earth?