A satellite is given a boost of 550 MJ of energy to move it from its initial orb

Question

A satellite is given a boost of 550 MJ of energy to move it from its initial orbit at an altitude of 150 km to a higher altitude orbit. If the satellite has a mass of 1.07 x103 kg, what is the new altitude it reaches? Take the mass of the Earth to be 5.97 x 1024 kg and its radius to be 6.371 x 105 m. 109.1 Consider the gravitational force between the satellite and Earth. How is this force related to the centripetal force that keeps the satellite in its circular orbit? What is the difference in energy of the system between the two orbits? km

Explanation / Answer

Centripetal force on a satellite of mass m moving at velocity v in an orbit of radius r is = mv2/r

But this is equal to the gravitational force (F) between the earth (mass M) and the satellite (mass m):

F =GMm/r² and

so mv2= GMm/r

But kinetic energy = 1/2*m*v2 and so:

kinetic energy of the satellite = 1/2*GMm/r

therefore total energy of a satellite in orbit of radius r is

-GMm/r + GMm/2r = -GMm/2r

total energy of satellite at height of 150 km = - GMm/2(R + 150000)

M = 5.97*1024 kg

R = 6.371 *106 m

m = 1.07*103 kg

G = 6.67×1011 N·(m/kg)2

if new altitude is x m

from conservation of energy

- GMm/2(R + 150000) + 550*106 = - GMm/2(R + x)

550*106 = (GMm/2) [ 1/(R + 150000) - 1/(R + x) ]

550*106 = (GMm/2) [ (R + x) - (R + 150000) ] / [(R + x)(R + 150000) ]

550*106 = (GMm/2) [x - 150000] / [(R + x)(R + 150000)]

(550*106)*2 = (6.67*10-11)*(5.97 *1024)*(1.07*103) *(x - 150000) / [(R + x)(R + 150000)]

(13.73*2)[(R + x)(R + 150000)] = 1010*[x - 150000]

27.46[R2 + 150000R + Rx + 150000x] = 1010(x) - (1.5)*1015

27.46[R2 + 150000R] + 1.5(10^15) = x[1010 -27.46R - 27.46(150000)]

9820933340x = (1.140833690860000 + 1.5)1015

x = 2.640833690860000(1015)/[9.820933340(109)

x = 10^6[0.2688984437] m = 268.898 km = 269 km

x = 269 km

Ans: the new altitude it reaches is ~= 269 km

Related Questions

Navigate

• Browse (All)
• Browse A
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.