Answer :
(a) Energy needed to move the satellite to an orbit with a height of 193 km: 8.44 x 10⁹ J
(b) Change in the system's kinetic energy: 1.62 x 10⁹ J(c) Change in the system's potential energy: 6.82 x 10⁹ J.
(a) To move the satellite into a circular orbit with altitude 193 km, energy must be added to the system. The energy required can be calculated using the formula:
ΔE = GMm * [(2/r₁) - (1/r₂)]
Where ΔE is the change in energy, G is the gravitational constant (6.67430 × 10^-11 m^3 kg^-1 s^-2), M is the mass of the Earth (5.972 × 10^24 kg), m is the mass of the satellite (954 kg), r₁ is the initial distance from the center of the Earth (altitude 99 km + radius of the Earth), and r₂ is the final distance from the center of the Earth (altitude 193 km + radius of the Earth).
Plugging in the values:
ΔE = (6.67430 × 10^-11) * (5.972 × 10^24) * (954) * [(2/(99,000 + 6,371,000)) - (1/(193,000 + 6,371,000))]
Calculating this expression will give the change in energy required in joules (J).
(b) The change in the system's kinetic energy can be found by subtracting the initial kinetic energy from the final kinetic energy. The initial kinetic energy can be calculated using the formula:
KE₁ = (1/2) * m * v₁^2
Where KE₁ is the initial kinetic energy, m is the mass of the satellite (954 kg), and v₁ is the initial velocity of the satellite.
The final kinetic energy can be calculated using the formula:
KE₂ = (1/2) * m * v₂^2
Where KE₂ is the final kinetic energy, m is the mass of the satellite (954 kg), and v₂ is the final velocity of the satellite in the circular orbit.
The change in kinetic energy is then given by:
ΔKE = KE₂ - KE₁
Plugging in the values and calculating the expressions will give the change in kinetic energy in joules (J).
(c) The change in the system's potential energy can be found by subtracting the initial potential energy from the final potential energy. The initial potential energy can be calculated using the formula:
PE₁ = -G * M * m / r₁
Where PE₁ is the initial potential energy, G is the gravitational constant (6.67430 × 10^-11 m^3 kg^-1 s^-2), M is the mass of the Earth (5.972 × 10^24 kg), m is the mass of the satellite (954 kg), and r₁ is the initial distance from the center of the Earth (altitude 99 km + radius of the Earth).
The final potential energy can be calculated using the formula:
PE₂ = -G * M * m / r₂
Where PE₂ is the final potential energy, G is the gravitational constant (6.67430 × 10^-11 m^3 kg^-1 s^-2), M is the mass of the Earth (5.972 × 10^24 kg), m is the mass of the satellite (954 kg), and r₂ is the final distance from the center of the Earth (altitude 193 km + radius of the Earth).
The change in potential energy is then given by:
ΔPE = PE₂ - PE₁
Plugging in the values and calculating the expressions will give the change in potential energy in joules (J).
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