Calculating Earth Displacement from Free Fall of 1kg Object

[gasar8]

Homework Statement

We release $m_o = 1$ kg object from h=1m height. How much does the Earth move (x)? I just need the comfirmation if I did correctly?

Homework Equations

Conservation of momentum:
$m_o v_o + m_E v_E = 0$

The Attempt at a Solution

I wrote the energy conservation law as (beginning = end):
$$m_o g h = \frac{1}{2} m_o v_o^2 + \frac{1}{2} m_E v_E^2,$$
and from momentum conservation got:
$$g h= \frac{1}{2} v_o^2 \bigg(1 + \frac{m_o}{m_E} \bigg),$$
and now, I insert the free fall relation $v_o = \sqrt{2g(h-x)}$ from where I extract $x$ as:
$$x = h\frac{\frac{m_o}{m_E}}{1+\frac{m_o}{m_E}} .$$
It seems reasonable, because it is something small.

 

[BvU]

I wrote the energy conservation law

Is your scenario describing a fully elastic collision ? If not, you'll need some other relationship...

 

[gasar8]

Doesn't say anything about that, so I would assume that it is elastic?

 

[BvU]

then the object bounces back up 1 m

 

[hutchphd]

Homework Statement

We release $m_o = 1$ kg object from h=1m height. How much does the Earth move (x)? I just need the comfirmation if I did correctly?

Homework Equations

Conservation of momentum:
$m_o v_o + m_E v_E = 0$

The Attempt at a Solution

I wrote the energy conservation law as (beginning = end):
$$m_o g h = \frac{1}{2} m_o v_o^2 + \frac{1}{2} m_E v_E^2,$$
and from momentum conservation got:
$$g h= \frac{1}{2} v_o^2 \bigg(1 + \frac{m_o}{m_E} \bigg),$$
and now, I insert the free fall relation $v_o = \sqrt{2g(h-x)}$ from where I extract $x$ as:
$$x = h\frac{\frac{m_o}{m_E}}{1+\frac{m_o}{m_E}} .$$
It seems reasonable, because it is something small.

To assure yourself further what limits can you look at? What if the two masses are equal? What if the object is much more massive than earth? Those limits give you reasonable answers so you can do this without any "expert" advice!

 

[gasar8]

Ok, that does not seem right. :) So I must change the momentum conservation also? I am not sure what exactly changes?

 

[hutchphd]

then the object bounces back up 1 m

I think you misunderstand the question

 

[BvU]

Enlighten me

 

[hutchphd]

The question is how far the Earth moves during the fall, I believe. This is independent of how elastic the colision.

 

[BvU]

I agree. What happens to the center of mass, my dear gasar ?

 

[gasar8]

During the fall, I supose it should stay at rest?