Quick question on Newton's third law & energy conservation

[DavidDoakes]

I have a quick question on Newton's third law.

When a 100kg astronaut in space is holding a 100 kg rock and then pushes it away from him with 50 joules of energy (the kinetic energy put into the system) and a second astronaut (observer) is watching, sitting still relatively to the location where the other astronaut was holding the rock and generating the 50 joules before the push, does the observer see the rock float to the left of him with 1 m/s and the astronaut float to the right of him with 1 m/s or does he see the rock float to the left with 0.707 m/s and the astronaut to the right with 0.707 m/s? In other words, is the total kinetic energy he observes equal to 100 joules or 50 joules?

 

[Dale]

Hi DavidDoakes, welcome to PF!

pushes it away from him with 50 joules of energy (the kinetic energy put into the system) ... is the total kinetic energy he observes equal to 100 joules or 50 joules?

Energy is conserved. What does that imply for your question?

 

[DavidDoakes]

Hi DavidDoakes, welcome to PF!Energy is conserved. What does that imply for your question?

I'd say both the astronaut and rock would have speeds of 0.707 m/s relative to the observer (otherwise you can could catch both the astronaut's and the rock's kinetic energy, transport it to the center of the system and use half of it to repeat the whole thing, gaining 50 joules of free energy every time), but there are so many (popular) sources out there that explain it like they would have speeds of 1 m/s relative to the observer, so I was wondering if I missed something.

 

[Dale]

I'd say both the astronaut and rock would have speeds of 0.707 m/s relative to the observer (otherwise you can could catch both the astronaut's and the rock's kinetic energy, transport it to the center of the system and use half of it to repeat the whole thing, gaining 50 joules of free energy every time)

You are correct, both in your conclusion and your reasoning.

If you look at a more detailed analysis, suppose the astronaut has an arm length of 1 m and pushes with 50 N force. If the astronaut were pushing on a very massive object that did not displace much then he would travel 1 m during the push and all 50 J would go into his KE. If the astronaut were pushing on a very light object so that he did not displace much then that object would travel 1 m during the push and all 50 J would go into the object. Since the mass is the same as the astronaut, neither of these ideal situations occur, instead both the astronaut and the rock move 0.5 m during the push, so each gets 25 J.

 

[DavidDoakes]

You are correct, both in your conclusion and your reasoning.

If you look at a more detailed analysis, suppose the astronaut has an arm length of 1 m and pushes with 50 N force. If the astronaut were pushing on a very massive object that did not displace much then he would travel 1 m during the push and all 50 J would go into his KE. If the astronaut were pushing on a very light object so that he did not displace much then that object would travel 1 m during the push and all 50 J would go into the object. Since the mass is the same as the astronaut, neither of these ideal situations occur, instead both the astronaut and the rock move 0.5 m during the push, so each gets 25 J.

Thank you for clearing this up for me. Doesn't it bother you that so many sources explain this incorrectly (or at least in such a way that it's easy to interpret it incorrectly)?