Newtonian gravity theory and energy

In summary, the conversation discusses the calculation of the impact speed of a 1.0kg object released from rest at 500km above the earth's surface, without considering air resistance. The calculation uses the equations for gravitational potential energy and kinetic energy, and takes into account the change in GPE between the initial and final positions. The error in the initial calculation is identified and corrected, leading to the correct answer of 3.02km/s. It is also noted that the GPE is zero at the surface of the earth, not at the center.
  • #1
Lord Anoobis
131
22

Homework Statement


A 1.0kg object is released form rest 500km above the earth. What is its impact speed as it hits the ground? Ignore air resistance.

Homework Equations


##U_g = \frac{GmM_e}{r}##
##K = \frac{1}{2}mv^2##
## \Delta U = - \Delta K##

The Attempt at a Solution


Using energy conservation I arrived at
##\frac{GmM_e}{r} = \frac{1}{2}mv^2##, then ##v = sqrt{\frac{2GM_e}{r}}##

Plugging in values, ##v = \sqrt{\frac{2(6.67\times10^{-11})(5.98\times10^{24})}{(6.37\times10^6 +5.00\times10^5)}} = 10776 m/s##, never mind the significant figures.
This is rather far from the correct answer though, which is 3.02km/s.
 
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  • #2
(According to your equation for the gravitational potential energy) is the GPE zero at the surface of Earth?

P.S.
There should be a negative sign on Ug
 
  • #3
Nathanael said:
(According to your equation for the gravitational potential energy) is the GPE zero at the surface of Earth?
Forgot to put the negative sign in there. By the look of it the GPE is zero at the centre of the earth, which accounts for the excess energy, correct?
 
  • #4
Solve for v
upload_2015-7-12_14-43-4.png
 
  • #5
Lord Anoobis said:
Forgot to put the negative sign in there. By the look of it the GPE is zero at the centre of the earth, which accounts for the excess energy, correct?
It has nothing to do with the center of the Earth.

You have two positions, 500km above the surface, and at the surface. The change in the GPE between these two positions is equal to the change in kinetic energy. Your calculation neglected the GPE at the surface of Earth.
(Your equation for the GPE takes the zero-potential to be at infinity, not at the surface.)
 
  • #6
Engineer at UIC said:
Solve for v
View attachment 85864
Nathanael said:
It has nothing to do with the center of the Earth.

You have two positions, 500km above the surface, and at the surface. The change in the GPE between these two positions is equal to the change in kinetic energy. Your calculation neglected the GPE at the surface of Earth.
(Your equation for the GPE takes the zero-potential to be at infinity, not at the surface.)
True. I got that mixed up. Problem solved and thanks for the input. I feel I need to have another closer look at this section though.
 

FAQ: Newtonian gravity theory and energy

1. What is Newton's Law of Universal Gravitation?

Newton's Law of Universal Gravitation states that every object in the universe attracts every other object with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.

2. How does Newton's Law of Universal Gravitation explain the motion of planets?

According to Newton's Law of Universal Gravitation, the gravitational force between the Sun and the planets causes them to orbit in elliptical paths. The gravitational force is strongest when the planets are closest to the Sun and weakest when they are farthest away, resulting in their orbital motion.

3. What is the difference between potential energy and kinetic energy in Newtonian gravity theory?

Potential energy in Newtonian gravity theory is the stored energy an object has due to its position in a gravitational field. Kinetic energy, on the other hand, is the energy an object possesses due to its motion. In the context of gravity, this is the energy an object has due to its speed and direction of motion in a gravitational field.

4. Can Newtonian gravity theory be applied to all objects in the universe?

Newtonian gravity theory can be applied to most objects in the universe, but it becomes less accurate for extremely massive or extremely small objects, or objects that are moving at very high speeds. In these cases, the theory of general relativity is a more accurate description of gravity.

5. How does Newtonian gravity theory relate to Einstein's theory of general relativity?

Newtonian gravity theory is a simplified version of Einstein's theory of general relativity. While Newtonian gravity theory assumes that gravity is a force acting between objects, Einstein's theory states that gravity is the curvature of space-time caused by massive objects. In most cases, Newtonian gravity theory is sufficient for practical applications, but Einstein's theory is needed for more precise calculations and for objects that are affected by strong gravitational fields.

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