Respuesta :
To solve this problem we will apply the concepts related to gravitational potential energy. The existing difference between the two heights will allow us to find the net energy to perform the action determined in the statement. The energy required to boost the shuttle to the new orbit is
[tex]\Delta E = E_2-E_1[/tex]
[tex]\Delta E = GMm(\frac{1}{2r_1}-\frac{1}{2r_2})[/tex]
Our values are
[tex]M = 5.972*10^{24}kg \rightarrow[/tex] Mass of Earth
[tex]R = 6.371*10^6m \rightarrow[/tex] Radius of Earth
[tex]G = 6.674*10^{-11}m^3/kgs^2 \rightarrow[/tex] Gravitational Universal Constant
[tex]m = 65000kg[/tex]
The initial radius of the orbit of the shuttle is
[tex]r_1 = R+h_1[/tex]
[tex]r_1 = 6.371*10^6+2.5*10^5[/tex]
[tex]r_1 = 6.621*10^6m[/tex]
Final Orbit radius of the shuttle is
[tex]r_2 = R+h_2[/tex]
[tex]r_2 = 6.371*10^6+6.1*10^5[/tex]
[tex]r_2 = 6.98*10^6m[/tex]
Replacing we have that
[tex]\Delta E = (6.674*10^{-11})(5.972*10^{24})(65000)(\frac{1}{2(6.621*10^6)}-\frac{1}{2(6.98*10^6)})[/tex]
[tex]\Delta E = 1.04*10^{11}J[/tex]
The energy required to boost the shuttle to the new orbit is; ΔE = 104 × 10⁹ J
What is the energy required?
We are given;
Mas of Earth; M = 5.924 * 10²⁴ kg
Radius of Earth; R = 6371 * 10³ m
Gravitational Constant; G = 6.67 * 10⁻¹¹ N.m²/kg²
Mass of space shuttle; m = 65000 kg
initial height of orbit; h₁ = 250 km = 250000 m
Final height of orbit; h₂ = 610 km = 610000 m
Thus;
Initial radius; r₁ = R + h₁
r₁ = (6371 * 10³) + 250000
r₁ = 662.1 * 10³ m
Final radius of orbit;
r₂ = R + h₂
r₂ = (6371 * 10³) + 610000
r₂ = 6981 * 10³ m
Formula for energy required to boost the shuttle to the new orbit is;
ΔE = ¹/₂GMm(¹/r₁ - ¹/r₂)
ΔE =¹/₂ * 6.67 * 10⁻¹¹ * 5.924 * 10²⁴ * 65000 * (¹/(662.1 * 10³) - ¹/(6981 * 10³))
ΔE = 104 × 10⁹ J
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