If a rock fell 20,000 meters to Earth Question

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In summary, the time it would take for a 5 kg rock to hit Earth with no air resistance or other forces is 20,000 meters.
  • #1
Jack981
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I was thinking about this question:

So if a rock fell 20,000 meters to Earth assuming no air resistance and no other gravitational forces from anything but Earth, how long would it take the rock to hit Earth? Keep in mind acceleration is changing through the gravity equation as the rock gets closer and closer to Earth -> g=m1(m2)G/r^2.

(I'm not sure if its necessary but if it is, say the rock is 5 kg).

I put undergrad because this was not a question that came up in class and I think it involves integration.

This is not a hw/cw question it is a question I came up with on my own.
 
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  • #2
Welcome to PF, Jack. Do you have any specific questions about the question you were thinking about?
 
  • #3
Hello, I wanted to know if anyone had a possible solution to the question which is how much time it would take to hit Earth. I've been trying to figure it out but can't figure out how to fit time into the overall equation because I always just get the equations in terms of radius.
 
  • #4
If you search PF, sometimes you can find similar inquiries. @HallsofIvy did a good development on this topic in this PF thread. I must admit I stall out on the subject of the elliptic integral. Maybe you can take it further.
 
  • #5
Am I missing something here? Why do you need an elliptic integral? Set up a free body diagram where the force found from Newton's law of gravity is equal to the mass of the ball times its acceleration. The equation is separable and so is fairly easily integrated (twice) to get the separation between the two bodies as a function of time. Maybe I am missing something here?
 
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  • #6
Jack981 said:
So if a rock fell 20,000 meters to Earth assuming no air resistance and no other gravitational forces from anything but Earth, how long would it take the rock to hit Earth? Keep in mind acceleration is changing through the gravity equation as the rock gets closer and closer to Earth -> g=m1(m2)G/r^2.
20 km? How much does g really change? Do you really need to bother with that? If you decide no, do you know how to do the problem then?
 
  • #7
russ_watters said:
20 km? How much does g really change? Do you really need to bother with that? If you decide no, do you know how to do the problem then?

The time difference between calculations using constant surface and variable gravity for ##20km## is only ##0.17s##, I believe.
 
  • #8
Jack981 said:
I was thinking about this question:

So if a rock fell 20,000 meters to Earth assuming no air resistance and no other gravitational forces from anything but Earth, how long would it take the rock to hit Earth? Keep in mind acceleration is changing through the gravity equation as the rock gets closer and closer to Earth -> g=m1(m2)G/r^2.

(I'm not sure if its necessary but if it is, say the rock is 5 kg).

I put undergrad because this was not a question that came up in class and I think it involves integration.

This is not a hw/cw question it is a question I came up with on my own.

https://www.physicsforums.com/threa...ith-varying-acceleration.866218/#post-5437928
 

1. How fast would the rock be falling at 20,000 meters?

Assuming the rock is in a vacuum, it would be falling at approximately 2,190 meters per second.

2. What would be the impact force of the rock hitting the ground?

The impact force would depend on the mass and velocity of the rock, but it would likely be a significant force due to the high velocity of the fall.

3. How long would it take for the rock to reach the ground?

Using the formula t = √(2d/g) where t is time, d is distance, and g is the acceleration due to gravity (9.8 m/s²), it would take approximately 63 seconds for the rock to reach the ground.

4. Would the rock reach terminal velocity at 20,000 meters?

No, terminal velocity is the maximum velocity an object can reach while falling, and it depends on air resistance. Since the air is thinner at higher altitudes, the rock would continue to accelerate until it reaches a lower altitude where the air is denser.

5. Would the rock survive the fall?

That would depend on the composition and strength of the rock, as well as the conditions of impact. Some rocks are more durable than others, but a fall from 20,000 meters would likely result in significant damage or destruction of the rock.

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