Orbits and Energy

r

Consider an object of mass m in a circular orbit a distance r from the center of the Earth.

Orbits and Energy

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Consider an object of mass m in a circular orbit a distance r from the center of the Earth.

Orbits and Energy

Consider an object of mass m in a circular orbit a distance r from the center of the Earth.

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Orbits and Energy

Consider an object of mass m in a circular orbit a distance r from the center of the Earth.

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Circular Orbit

Orbits and Energy

Consider an object of mass m in a circular orbit a distance r from the center of the Earth.

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Circular Orbit

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Total Mechanical Energy

Orbits and Energy

Two small spaceships, each with a mass m = 2000. kg, are in the circular Earth orbit of the diagram, at altitude h of 400. km. Igor, the commander of one of the ships, arrives at

any fixed point in the orbit 90. s ahead of Sally, the commander of the other ship.

a. What are the ships’ orbital periods?

32

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Kepler’s Third

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Orbits and Energy

Two small spaceships, each with a mass m = 2000. kg, are in the circular Earth orbit of the diagram, at altitude h of 400. km. Igor, the commander of one of the ships, arrives at

any fixed point in the orbit 90. s ahead of Sally, the commander of the other ship.

a. What are the ships’ orbital periods?

32

2 4r

GMT

Kepler’s Third

GMr

T3

2

kg

kg

Nm

mT

242

211

36

1098.51067.6

1077.62

sT 5540

P

Orbits and Energy

Two small spaceships, each with a mass m = 2000. kg, are in the circular Earth orbit of the diagram, at altitude h of 400. km. Igor, the commander of one of the ships, arrives at

any fixed point in the orbit 90. s ahead of Sally, the commander of the other ship.

b. What are the ships’ orbital speeds?

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Gvo

m

kg

kg

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24

2

211

1077.6

1098.51067.6

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For a circular orbit

Orbits and Energy

Two small spaceships, each with a mass m = 2000. kg, are in the circular Earth orbit of the diagram, at altitude h of 400. km. Igor, the commander of one of the ships, arrives at

any fixed point in the orbit 90. s ahead of Sally, the commander of the other ship.

b. What are the ships’ orbital speeds?

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vo 2

s

mvo 5540

1077.62 6

smvo 7680

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speed, circular orbit

OR

Orbits and Energy

Two small spaceships, each with a mass m = 2000. kg, are in the circular Earth orbit of the diagram, at altitude h of 400. km. Igor, the commander of one of the ships, arrives at

any fixed point in the orbit 90. s ahead of Sally, the commander of the other ship.

c. What is the total mechanical energy of either ship?

rMm

GET 2 For a circular orbit

m

kgkg

kg

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24

2

211

1077.62

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Orbits and Energy

Two small spaceships, each with a mass m = 2000. kg, are in the circular Earth orbit of the diagram, at altitude h of 400. km. Igor, the commander of one of the ships, arrives at

any fixed point in the orbit 90. s ahead of Sally, the commander of the other ship.

c. What is the total mechanical energy of either ship?

gkT EEE

m

kgkg

kg

Nm

smkgET

6

24

2

211

2

1077.6

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7680.200021

JET101089.5

rMm

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21

OR

Orbits and Energy

At point P in the diagram, Sally fires an instantaneous burst in the forward direction, reducing her ship’s speed by 1.00 %.

d. What is her ship’s new kinetic energy and potential energy immediately after this burst?

2

21

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Orbits and Energy

At point P in the diagram, Sally fires an instantaneous burst in the forward direction, reducing her ship’s speed by 1.00 %.

d. What is her ship’s new kinetic energy and potential energy immediately after this burst?

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Orbits and Energy

At point P in the diagram, Sally fires an instantaneous burst in the forward direction, reducing her ship’s speed by 1.00 %.

e. What is her ship’s new total mechanical energy immediately after this burst?

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Notice that Sally’s total mechanical energy is less than before.

(More Negative!)

Orbits and Energy

At point P in the diagram, Sally fires an instantaneous burst in the forward direction, reducing her ship’s speed by 1.00 %.

f. What is her ship’s new semimajor axis immediately after this burst?

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Orbits and Energy

At point P in the diagram, Sally fires an instantaneous burst in the forward direction, reducing her ship’s speed by 1.00 %.

g. Which of the dashed elliptical orbits shown in the figure will Sally’s ship then take?

P1

2

Notice that the semimajor axis is less than the original r!

Orbits and Energy

At point P in the diagram, Sally fires an instantaneous burst in the forward direction, reducing her ship’s speed by 1.00 %.

g. Which of the dashed elliptical orbits shown in the figure will Sally’s ship then take?

P

Notice that the semimajor axis is less than the original r!

Orbits and Energy

At point P in the diagram, Sally fires an instantaneous burst in the forward direction, reducing her ship’s speed by 1.00 %.

h. What will Sally’s new orbital period be?

32

2 4a

GMT

Kepler’s Third

GMa

T3

2

kg

kg

Nm

mT

242

211

36

1098.51067.6

1063.62

sT 5370

Orbits and Energy

i. When Sally returns to point P she fires an instantaneous burst in the backward direction returning her ship to its original speed. Is Sally now ahead or behind of Igor and by how much?

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s 170

But Sally was originally 90. s behind so

ahead! sss .80.90170

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Orbits and Energy

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