Energy of Earth Orbiting Around the Sun

In summary: As for the solution, it looks mostly fine, except for the average distance part. You could also make it more clear that you are using the energy conservation law to get the KE at the extremes.In summary, the Earth's distance from the sun varies from 1.471 x 108 km to 1.521 x 108 km during the year. To determine the difference in (a) the potential energy, (b) the earth's kinetic energy, and (c) the total energy between these extreme points, one must use the centripetal acceleration formula and the law of conservation of energy. However, calculating an average distance between the two extremes may not be helpful in finding the solution. Instead, finding the potential
  • #1
Ishaan S
19
0

Homework Statement


The Earth's distance from the sun varies from 1.471 x 108 km to 1.521 x 108 km during the year. Determine the difference in (a) the potential energy, (b) the earth's kinetic energy, and (c) the total energy between these extreme points. Take the sun to be at rest.

Homework Equations


FGrav = (GMm)/r2

U = integral(a→b)(GMm/r2)dr

KE = .5mv2

Law of the Conservation of energy: Ei = Ef

ac = v2 / r

The Attempt at a Solution



I got (a) easily. I need help with b and c, though.

I tried taking the average distance from the sun, which is just the length of the semi-major axis, which is

1.521 x 108 km - 1.471 x 108 km = 5.00 x 109 m = rave

Then I used the centripetal acceleration formula to get

Fa = FG = (GMsun / (rave)2) = (v2)ave/rave.

Solving for vave I get

vave = 2.7 x 1010 m/s.

I then calculated average kinetic energy to be

.5 * mearth * (vave)2 = 2.1 * 1045 J.
Average gravitational energy would be

-GMm/rave = -1.6 * 1035 J.

Average total energy would be average kinetic + average potential.

Since energy is constant the average energy is constant for instantaneous energy.

Therefore, at perihelion and aphelion, I would have average total energy - potential energy at the point, correct?

Any help would be appreciated, thanks.
 
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  • #2
Ishaan S said:

Homework Statement


The Earth's distance from the sun varies from 1.471 x 108 km to 1.521 x 108 km during the year. Determine the difference in (a) the potential energy, (b) the earth's kinetic energy, and (c) the total energy between these extreme points. Take the sun to be at rest.

Homework Equations


FGrav = (GMm)/r2

U = integral(a→b)(GMm/r2)dr

KE = .5mv2

Law of the Conservation of energy: Ei = Ef

ac = v2 / r

The Attempt at a Solution



I got (a) easily. I need help with b and c, though.

I tried taking the average distance from the sun, which is just the length of the semi-major axis, which is

1.521 x 108 km - 1.471 x 108 km = 5.00 x 109 m = rave

This calculation makes no sense. How can you subtract two positive numbers and have the difference be greater than either number?

Also, it's not even how you calculate an average distance.

Then I used the centripetal acceleration formula to get

Fa = FG = (GMsun / (rave)2) = (v2)ave/rave.

Solving for vave I get

vave = 2.7 x 1010 m/s.

Really? What's the speed of light in a vacuum? Do you have the Earth going faster or slower than light itself?

http://en.wikipedia.org/wiki/Speed_of_light

I then calculated average kinetic energy to be

.5 * mearth * (vave)2 = 2.1 * 1045 J.
Average gravitational energy would be

-GMm/rave = -1.6 * 1035 J.

Average total energy would be average kinetic + average potential.

Since energy is constant the average energy is constant for instantaneous energy.

Therefore, at perihelion and aphelion, I would have average total energy - potential energy at the point, correct?

Any help would be appreciated, thanks.

Correct your mistakes as pointed out above.
 
  • #3
OK.

The mean distance is (1.521*108 km + 1.471*108 km)/2 = 1.496*108 km = 1.496*1011 m.

Then the average velocity comes from the centripetal acceleration formula:

Fa = FG = GM/(rave)2 = (vave)2 / rave

Solve for vave

vave = 30,000 m/s.

So average KE = ,5mvave2 = 2.7*1033 J.

Average potential energy is

-GMm/rave = -5.3 * 1033 J.

So total energy is U + KE = -2.7*1033 J.

So at perihelion the KE would be

Etotal - (-GMm/rclosest distance) = 2.8*1033 J.

At aphelion the KE would be

Etotal - (-GMm/rfar distance) = 2.6*1033 J.

Is this correct?
 
  • #4
I do not understand how an 'average' distance could be helpful in answering the question. Just find the PE and KE at the given extremes.
Further, it is not clear what would be meant by an average distance from the sun. Taking the average of the two extreme distances is unlikely to produce anything useful. An integral would make more sense, but even then there are choices to be made. You could take an average over time, or an average over theta, etc.
 

Related to Energy of Earth Orbiting Around the Sun

1. How does the Earth's orbit around the Sun affect its energy?

The Earth's orbit around the Sun is a result of the gravitational pull between the two bodies. As the Earth moves along its elliptical orbit, it experiences changes in its distance from the Sun, resulting in variations in its potential and kinetic energy.

2. What is the source of the Earth's energy as it orbits the Sun?

The primary source of the Earth's energy as it orbits the Sun is the Sun itself. The Sun's immense gravitational force pulls the Earth towards it, causing the Earth to move in its orbit. The Sun's radiation also provides energy to sustain life on Earth.

3. How does the Earth's distance from the Sun affect its energy?

The Earth's distance from the Sun directly affects its potential and kinetic energy. When the Earth is closer to the Sun, it experiences a stronger gravitational pull, resulting in higher potential energy. As the Earth moves farther away from the Sun, its potential energy decreases and its kinetic energy increases.

4. Does the Earth's orbit around the Sun ever change?

Yes, the Earth's orbit around the Sun is constantly changing due to various factors such as gravitational interactions with other planets, the Sun's own movement, and the Earth's axial tilt. However, these changes are relatively minor and do not significantly affect the Earth's overall energy.

5. Can the Earth's orbit around the Sun affect its climate and seasons?

Yes, the Earth's orbit around the Sun plays a crucial role in determining its climate and seasons. The tilt of the Earth's axis and its elliptical orbit contribute to the changing of seasons and the varying amounts of solar radiation received by different parts of the Earth, resulting in different climates.

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