Math  /  Calculus

QuestionA 4025 kg spacecraft is in circular orbit around the Earth at an altitude of 256 km . Determine; a) the kinetic energy of the spacecraft b) the gravitational potential of the spacecraft 42.56×105 m42.56 \times 10^{5} \mathrm{~m} c) the total orbital energy of the space craft d) the binding energy of the space craft e) the speed required for escape at this height f) the work required to move the spacecraft to a new orbit of altitude 628 km g) homework; pg WS \#'s 4, 5, 7, 8, \& 10

Studdy Solution

STEP 1

1. The spacecraft is in a stable circular orbit around the Earth.
2. The mass of the spacecraft is 4025 4025 kg.
3. The altitude of the spacecraft is 256 256 km above the Earth's surface.
4. The radius of the Earth is approximately 6371 6371 km.
5. The gravitational constant G=6.674×1011m3/kgs2 G = 6.674 \times 10^{-11} \, \text{m}^3/\text{kg} \cdot \text{s}^2 .
6. The mass of the Earth M=5.972×1024kg M = 5.972 \times 10^{24} \, \text{kg} .

STEP 2

1. Calculate the kinetic energy of the spacecraft.
2. Calculate the gravitational potential energy of the spacecraft.
3. Calculate the total orbital energy of the spacecraft.
4. Calculate the binding energy of the spacecraft.
5. Calculate the speed required for escape at this height.
6. Calculate the work required to move the spacecraft to a new orbit of altitude 628 km.

STEP 3

Calculate the kinetic energy of the spacecraft.
1.1 Determine the orbital radius r r : r=Earth’s radius+altitude=6371km+256km=6627km=6627000m r = \text{Earth's radius} + \text{altitude} = 6371 \, \text{km} + 256 \, \text{km} = 6627 \, \text{km} = 6627000 \, \text{m}
1.2 Use the formula for orbital speed v v : v=GMr v = \sqrt{\frac{GM}{r}}
1.3 Calculate the kinetic energy KE KE : KE=12mv2 KE = \frac{1}{2} m v^2

STEP 4

Calculate the gravitational potential energy of the spacecraft.
2.1 Use the formula for gravitational potential energy U U : U=GMmr U = -\frac{GMm}{r}

STEP 5

Calculate the total orbital energy of the spacecraft.
3.1 Use the formula for total energy E E : E=KE+U E = KE + U

STEP 6

Calculate the binding energy of the spacecraft.
4.1 Use the formula for binding energy BE BE : BE=E BE = -E

STEP 7

Calculate the speed required for escape at this height.
5.1 Use the formula for escape velocity ve v_e : ve=2GMr v_e = \sqrt{\frac{2GM}{r}}

STEP 8

Calculate the work required to move the spacecraft to a new orbit of altitude 628 km.
6.1 Determine the new orbital radius r r' : r=Earth’s radius+new altitude=6371km+628km=6999km=6999000m r' = \text{Earth's radius} + \text{new altitude} = 6371 \, \text{km} + 628 \, \text{km} = 6999 \, \text{km} = 6999000 \, \text{m}
6.2 Calculate the change in gravitational potential energy: ΔU=(GMmr)(GMmr) \Delta U = \left(-\frac{GMm}{r'}\right) - \left(-\frac{GMm}{r}\right)
6.3 Calculate the work done W W : W=ΔU W = \Delta U
Note: The homework part is not a calculation step and is not addressed here.

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