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a 991-kg satellite orbits the earth at a constant altitude of 95-km. how much energy must be added to the system to move the satellite into a circular orbit with altitude 194 km?

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Final answer:

To calculate the energy needed to move a satellite from one orbit to another, determine the difference in mechanical energy between the two orbits. This involves using the formulas for gravitational potential energy and kinetic energy, adjusting for the altitudes above Earth's surface.

Step-by-step explanation:

The question is asking how much energy is needed to move a satellite from a lower to a higher orbit around the Earth.

This involves calculating the change in gravitational potential energy and kinetic energy of the orbiting satellite.

To find the additional energy required, we'd need to calculate the difference in total mechanical energy (potential + kinetic) between the two orbital heights.

We consider the two orbits as circular and we use the conservation of mechanical energy principle alongside formulae for gravitational potential energy (-G * M * m / r) and kinetic energy (1/2 * m * v^2), where G is the gravitational constant, M is the mass of Earth, m is the mass of the satellite, r is the distance from the Earth's center to the satellite, and v is the orbital velocity.

To solve this problem, determine the satellite's initial potential and kinetic energy at 95 km altitude, then do the same for the final orbital altitude of 194 km.

The difference between the final energy and the initial energy will give you the amount of energy that needs to be added to the system. To get the exact distances from the Earth's center, you would need to add the Earth's radius (~6371 km) to each altitude.

User Ashish Emmanuel
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The energy that must be added to the system to move the satellite into a higher altitude is 9 x 10⁸ J.

How to calculate the energy that must be added?

The gravitational potential energy of the satellite in the different position is calculated by applying the following formula as shown below.

U = - (GMm)/r

where;

  • G is the universal gravitation constant
  • M is the mass of earth
  • m is the mass of the satellite
  • r is the position of the satellite from center of earth

When the satellite it at the altitude of 95 km;

r = R(earth) + 95 km

r = 6371 km + 95 km

r = 6,466 km

The initial potential energy of the satellite is;

U₁ = -(6.67 x 10⁻¹¹ x 5.97 x 10²⁴ x 991 ) / (6,466,000)

U₁ = - 6.1 x 10¹⁰ J

When the satellite moved to the altitude of 194 km;

r₂ = 6371 km + 194 km

r₂ = 6,565 km

The final potential energy of the satellite is;

U₂ = - (6.67 x 10⁻¹¹ x 5.97 x 10²⁴ x 991 ) / (6,565,000)

U₂ = - 6.01 x 10¹⁰ J

The difference in the energy;

ΔU = U₂ - U₁

ΔU = - 6.01 x 10¹⁰ J - (- 6.1 x 10¹⁰ J)

ΔU = - 6.01 x 10¹⁰ J + 6.1 x 10¹⁰ J

ΔU = 9 x 10⁸ J

User Tanisha
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