Momentum of Astronaut Throwing Apples in Space

[edowuks]

Homework Statement

Astronaut is at rest in space. He has back of apples with him. How should he throw the rocks away to gain maxium velocity. (one by one, whole back, slice the apples into smaller fracments, group the apples, etc)

Homework Equations

p=mv

The Attempt at a Solution

I marked M_tot="total mass of apples"=n*m_app where m_app is mass of one apple and n the quantity of apples. When he throws the whole back (ignoring the mass of back) he gains velocity V_a=(M_tot*V)/(m_ast-M_tot) where V is the velocity he throws the back and m_ast is his mass. Now when he throws one by one he is at rest first so after throwing the momentum must be p_1=(m_ast-m_app)*V_ast1=m_appV, now when he throws second apple the momentum of system is P_1 so new momentum is P_2=(m_ast-2*m_app)V_ast2=2*m_app*V and continuing this way I get exact same velocity that I got at first since P_n=V_astn*(m_ast-M_tot)=m_tot*V.
But does it really go this easy way (I thought I needed to integrate something, but couldn,t get any equation for force), and am I thinking that velocity right. Now it is velocity relative to astronaut, should it be velocity relative to space?

Thanks for your help, and sorry my not-so-good english skills

 

[Perillux]

yes you appear to have done it correctly. and your result is correct too, you should get the same result no matter how small the pieces are as long as you throw each piece with the same velocity.

there is no universal reference frame. So in this problem we would just measure their velocities relative to an observer at rest.

 

[edowuks]

I rethought it and am I really right. I mean in that solution I have assumed that the velocity astronaut gives to one apple is same as the velocity he gives to whole bag. But is the velocity really constant, shouldn't it be impulse that is constant since astronaut is human he has maxium force he can give to apple. But thinking that way he recaives much higher velocity by throwing apples one by one, that throwing whole bag.

 

[Perillux]

excellent point. I think the question is a little vague in this area, so it's difficult to tell.
If we assume he could throw all pieces with the exact same velocity (which is probably unlikely) then you did it correctly.

So, a better way to do it would probably be to assume the astronaut can only apply a certain force to each piece (which is more realistic). In this case each piece would have the same acceleration but different velocities.

But, it really all depends on how he throws it. If he throws the large bag but applies an impulse then his momentum will be greater. So the question is poorly worded.