Help on physics problem:distance Earth moves towards the apple

[Coco12]

Homework Statement

When you drop a 400g apple, Earth exerts a force that accelerates it at -9.80m/s^2.
The mass of the Earth is 5.98*10^24kg.
The acceleration of the Earth as it falls towards the apple is 6.56 • 10^-25m/s^2.
if the apple falls from a height of 5.00m, find the distance the Earth moves.

Homework Equations

vf^2=vi^2+2ad
t=2d/vi+vf

The Attempt at a Solution

I found the final velocity(9.90m/s) of the apple and the time it took (1.01s).
But how do I find the distance the Earth moves with only its acceleration?

 

[superdave]

You have time, acceleration, and you can infer initial velocity. Look at your other kinematic equations.

 

[Coco12]

so can I use the formula d=vit+1/2at^2?
will the intital velocity be 0 or 9.90m/s?

 

[superdave]

The initial velocity is 0 m/s.

 

[Nickie]

I would approach this problem by first understanding the forces at play. In this scenario, we have the force of gravity acting on both the apple and the Earth. The acceleration of the apple is due to the force of the Earth pulling on it, but the Earth also experiences a force from the apple. This is known as Newton's third law of motion: for every action, there is an equal and opposite reaction.

To find the distance the Earth moves towards the apple, we can use the equation F=ma, where F is the force, m is the mass, and a is the acceleration. In this case, we can use the mass and acceleration of the Earth to calculate the force it experiences from the apple.

F = (5.98*10^24kg) * (6.56 * 10^-25m/s^2) = 3.93 * 10^0N

Now that we have the force, we can use the equation F=ma again to find the distance the Earth moves. However, we need to know the mass of the Earth in this situation. Since the apple has a mass of 400g, we can assume that the Earth's mass is essentially unchanged. Thus, we can use the mass of the apple (0.4kg) to represent the mass of the Earth in this scenario.

F = (0.4kg) * a

3.93 * 10^0N = (0.4kg) * a

a = 9.82m/s^2

Now, we can use the equation vf^2=vi^2+2ad to find the distance the Earth moves. Since the initial velocity of the Earth is 0 (it is stationary), we can ignore the vi^2 term.

vf^2 = 2ad

(0m/s)^2 = 2 * (9.82m/s^2) * d

d = 0m

This result may seem counterintuitive, but it makes sense when we consider the relative masses of the Earth and the apple. The Earth is approximately 1.5 x 10^24 times more massive than the apple, so the force it experiences from the apple is incredibly small. Therefore, the distance it moves towards the apple is negligible.

In conclusion, the distance the Earth moves towards the apple is essentially 0 meters. This highlights the importance of considering all factors and understanding the