Find velocity of object dropped through hole in center of earth

In summary, the task was to find the velocity of an object dropped through a hole in the center of the Earth. The first-order differential equation for the object's velocity was derived, and the speed it achieves at the center of the Earth was determined using Gauss' Law. The result was then compared with the result obtained from conservation of energy considerations. The Earth's mass density was assumed to be uniform throughout. The task was successfully solved.
  • #1
maddogtheman
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[SOLVED] Find velocity of object dropped through hole in center of earth

1. Homework Statement
Suppose that a hole has been drilled through the center of the Earth, and that an object is dropped into this hole. Write a first-order differential equation for the object's velocity, v as a function of the distance r from the Earth's center (i.e., an equation involving dv/dr), and solve it to determine the speed the object achieves as it reaches the center of the Earth. Check this speed with the result you get from simple conservation of energy considerations. Consider the Earth's mass density to be uniform throughout. [Hint: recall Gauss' Law as it applies to the gravitational field of a spherically symmetric mass distribution.]
 
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Welcome to PF,

What have you attempted thus far?
 
  • #3
Solved it thanks
 

Related to Find velocity of object dropped through hole in center of earth

1. How does the velocity of an object dropped through a hole in the center of the Earth change?

The velocity of an object dropped through a hole in the center of the Earth changes due to the pull of gravity. As the object gets closer to the center of the Earth, the gravitational force increases, causing the velocity to increase as well.

2. Can the velocity of an object dropped through a hole in the center of the Earth ever be zero?

Yes, the velocity of an object dropped through a hole in the center of the Earth will be zero at the moment it reaches the center. This is because the gravitational force will be pulling the object equally in all directions, resulting in a net force of zero.

3. How does the mass of the object affect its velocity when dropped through a hole in the center of the Earth?

The mass of the object does not affect its velocity when dropped through a hole in the center of the Earth. The acceleration due to gravity is constant, so all objects will have the same velocity at the same distance from the center regardless of their mass.

4. What is the formula for calculating the velocity of an object dropped through a hole in the center of the Earth?

The formula for calculating the velocity of an object dropped through a hole in the center of the Earth is V = √(2GM/R), where G is the gravitational constant, M is the mass of the Earth, and R is the distance from the center of the Earth to the object.

5. Is it possible for the velocity of an object dropped through a hole in the center of the Earth to exceed the speed of light?

No, it is not possible for the velocity of an object dropped through a hole in the center of the Earth to exceed the speed of light. According to the theory of relativity, nothing can travel faster than the speed of light, which is approximately 299,792,458 meters per second.

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