# Calculating Earth Displacement from Free Fall of 1kg Object

• gasar8
In summary, the conversation discusses releasing a 1 kg object from a 1m height and determining how much the Earth moves during the fall using conservation of momentum and energy. The attempt at solving the problem involves writing the energy conservation law and using the free fall relation to extract the distance traveled by the Earth. There is also a discussion about the limits of the solution and the concept of center of mass.
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.

gasar8 said:
I wrote the energy conservation law
Is your scenario describing a fully elastic collision ? If not, you'll need some other relationship...

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

then the object bounces back up 1 m

gasar8 said:

## 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?

BvU said:
then the object bounces back up 1 m
I think you misunderstand the question

Enlighten me

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

gasar8
I agree. What happens to the center of mass, my dear gasar ?

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

## 1. What is free fall?

Free fall is the motion of an object falling under the influence of gravity alone, with no other external forces acting on it. It is a type of accelerated motion where the only force acting on the object is the force of gravity.

## 2. How is free fall related to earth displacement?

Earth displacement, or the movement of the Earth's surface, is not directly related to free fall. However, the force of gravity that causes free fall is the same force that keeps the Earth in its orbit around the sun and causes the Earth's surface to move in response to the gravitational pull of other objects, such as the moon.

## 3. What is the acceleration of free fall on Earth?

The acceleration of free fall on Earth is approximately 9.8 meters per second squared. This means that for every second an object is in free fall, its velocity increases by 9.8 meters per second.

## 4. How does air resistance affect free fall?

Air resistance, or the force of air pushing against an object as it falls, can affect the speed and motion of an object in free fall. In some cases, air resistance can slow down the object's acceleration, while in others it may cause the object to reach a terminal velocity, where it no longer accelerates due to the balance of forces.

## 5. Can an object experience free fall if it is not in a vacuum?

Yes, an object can still experience free fall in a non-vacuum environment, such as on Earth. However, the presence of air resistance may affect the object's motion and acceleration in free fall. In a vacuum, where there is no air resistance, an object will fall with a constant acceleration due to gravity.

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