- #1

Ishaan S

- 19

- 0

## Homework Statement

The Earth's distance from the sun varies from 1.471 x 10

^{8}km to 1.521 x 10

^{8}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

F

_{Grav}= (GMm)/r

^{2}

U = integral(a→b)(GMm/r

^{2})dr

KE = .5mv

^{2}

Law of the Conservation of energy: E

_{i}= E

_{f}

a

_{c}= v

^{2}/ 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 10

^{8}km - 1.471 x 10

^{8}km = 5.00 x 10

^{9}m = r

_{ave}

Then I used the centripetal acceleration formula to get

F

_{a}= F

_{G}= (GM

_{sun}/ (r

_{ave})

^{2}) = (v

^{2})

_{ave}/r

_{ave}.

Solving for v

_{ave}I get

v

_{ave}= 2.7 x 10

^{10}m/s.

I then calculated average kinetic energy to be

.5 * m

_{earth}* (v

_{ave})

^{2}= 2.1 * 10

^{45}J.

Average gravitational energy would be

-GMm/r

_{ave}= -1.6 * 10

^{35}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.