The energy required to move the satellite from one orbit to another can be expressed as the difference in the gravitational potential energy (GPE) between the two orbits. The GPE is given by:
GPE = -GMm / r,
where G is the gravitational constant, M is the mass of the Earth, m is the mass of the satellite, and r is the distance from the center of the Earth.
Initially, the satellite is at an altitude of 150 km, so its initial distance from the center of the Earth is the sum of the Earth's radius and the altitude:
r1 = 6.371 x 10^6 m + 150 x 10^3 m = 6.521 x 10^6 m.
Let the new altitude be h, then the distance from the center of the Earth in the new orbit is:
r2 = 6.371 x 10^6 m + h.
The energy difference required for the satellite to move to the new orbit is given as 540 MJ or 540 x 10^6 J. Therefore, we can write:
ΔGPE = GPE2 - GPE1 = 540 x 10^6 J.
Now we can substitute the expressions for GPE and solve for h:
- GMm / r2 + GMm / r1 = 540 x 10^6 J.
We can cancel out the satellite mass m and gravitational constant G, and rearrange the equation to solve for r2:
r2 = 1 / [(1 / r1) - (540 x 10^6 J) / (5.97 x 10^24 kg * 6.674 x 10^(-11) m^3/kg/s^2) ],
where we have substituted the values for Earth's mass M and gravitational constant G.
Now we can calculate the result:
r2 = 1 / [(1 / (6.521 x 10^6 m)) - (540 x 10^6 J) / (5.97 x 10^24 kg * 6.674 x 10^(-11) m^3/kg/s^2)]
=> r2 ≈ 7.00 x 10^6 m.
Finally, we can find the new altitude h by subtracting the Earth's radius:
h = r2 - 6.371 x 10^6 m = 7.00 x 10^6 m - 6.371 x 10^6 m = 0.629 x 10^6 m ≈ 629 km.
So the new altitude reached by the satellite after the energy boost is approximately 629 km.
A satellite is given a boost of 540 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 1.16 ✕ 10^3 kg, what is the new altitude it reaches? Take the mass of the Earth to be 5.97 x 10^24 kg and its radius to be 6.371 x 10^6 m.
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