Question

A space shuttle is initially in a circular orbit at a radius of \(r=6.60 \cdot 10^{6} \mathrm{~m}\) from the center of the Earth. A retrorocket is fired forward, reducing the total energy of the space shuttle by \(10 \%\) (that is, increasing the magnitude of the negative total energy by \(10 \%\) ), and the space shuttle moves to a new circular orbit with a radius that is smaller than \(r\). Find the speed of the space shuttle (a) before and (b) after the retrorocket is fired.

Short answer

Answer: The final circular orbit radius is \(r_\text{final} = \dfrac{6.6 \times 10^6 \text{ m}}{1.1}\). The speed of the space shuttle before the retrorocket firing is \(v = \sqrt{\dfrac{GM}{6.6 \times 10^6 \text{ m}}}\), and the speed after the retrorocket firing is \(v_\text{final} = \sqrt{\dfrac{GM}{\dfrac{6.6 \times 10^6 \text{ m}}{1.1}}}\).

Step-by-step solution

Write down the initial total mechanical energy

The total mechanical energy E of an object in a circular orbit is given by \(E = -\dfrac{GMm}{2r}\), where G is the gravitational constant, M is the mass of the Earth, m is the mass of the space shuttle, and r is the radius of the circular orbit. In this case, we have r = \(6.6 \times 10^6\) m.

Determine the final mechanical energy after the retrorocket firing

The problem states that the total energy of the space shuttle increases by 10%, so the final total mechanical energy will be \(E_\text{final} = 1.10 E\). We can express this in terms of the initial energy: \(E_\text{final} = -\dfrac{1.1GMm}{2r}\).

Find the final circular orbit radius

To find the radius of the new circular orbit, we can equate the final total mechanical energy with the new circular orbit energy: \( -\dfrac{1.1GMm}{2r} = -\dfrac{GMm}{2r_\text{final}}\), where \(r_\text{final}\) is the radius of the new circular orbit. Solving for \(r_\text{final}\), we get \( r_\text{final} = \dfrac{r}{1.1}\).

Calculate the speed of the space shuttle before the retrorocket firing

The speed \(v\) of the space shuttle in a circular orbit is given by the equation \(v = \sqrt{\dfrac{GM}{r}}\). We can plug in the values for G, M, and the initial radius r to find the initial speed of the space shuttle.

Calculate the speed of the space shuttle after the retrorocket firing

Similarly, the speed of the space shuttle in the new circular orbit is given by the equation \(v_\text{final} = \sqrt{\dfrac{GM}{r_\text{final}}}\). We can plug in the values for G, M, and the final radius \(r_\text{final}\) to find the final speed of the space shuttle. Solution:

Write down the initial total mechanical energy

\(E = -\dfrac{GMm}{2r} = -\dfrac{GMm}{2(6.6 \times 10^6 \text{ m})}\)

Determine the final mechanical energy after the retrorocket firing

\(E_\text{final} = 1.10 E = -\dfrac{1.1GMm}{2(6.6 \times 10^6 \text{ m})}\)

Find the final circular orbit radius

\(r_\text{final} = \dfrac{r}{1.1} = \dfrac{6.6 \times 10^6 \text{ m}}{1.1}\)

Calculate the speed of the space shuttle before the retrorocket firing

\(v = \sqrt{\dfrac{GM}{r}} = \sqrt{\dfrac{GM}{6.6 \times 10^6 \text{ m}}}\)

Calculate the speed of the space shuttle after the retrorocket firing

\(v_\text{final} = \sqrt{\dfrac{GM}{r_\text{final}}} = \sqrt{\dfrac{GM}{\dfrac{6.6 \times 10^6 \text{ m}}{1.1}}}\)

Key concepts

Circular Orbit

Understanding how a space shuttle orbits Earth involves grasping the concept of a circular orbit. A circular orbit is a path a spacecraft follows around a celestial body, such as the Earth, that creates a circular pattern. This happens when the gravitational pull of the Earth is exactly balanced by the shuttle's inertia, which tends to move it in a straight line. The combination of these two forces creates a stable loop, allowing the shuttle to keep orbiting at a constant distance from the Earth's center.

Gravitational attraction is the centripetal force that keeps the shuttle moving in a circle. The velocity of the shuttle in this orbit is crucial; it has to be just right. If it's too fast, the shuttle would fly off into space. If it's too slow, it would be pulled back towards Earth. With a perfect balance, the shuttle maintains a circular velocity, allowing it to revolve around Earth without getting closer or farther away.

Gravitational Force

The gravitational force is a fundamental aspect of space shuttle orbital mechanics. This invisible force attracts two objects with mass towards each other. In the context of a space shuttle orbiting Earth, this force is what keeps the shuttle from flying off into space.

Isaac Newton's law of universal gravitation expresses this force mathematically as proportional to the product of the two masses and inversely proportional to the square of the distance between their centers. This means that the closer a spacecraft is to Earth or the greater the Earth's mass, the stronger the gravitational pull. Equations comprising the Earth's gravitational constant and the masses of the Earth and the shuttle demonstrate this relationship and are crucial in calculating orbital trajectory and speed.

Retrorocket Firing

Retrorocket firing is a maneuver used to alter the orbit of a space shuttle. Retrorockets are engines capable of providing a thrust in the direction opposite to the shuttle's current path of motion. When these rockets are fired, the shuttle slows down, which might seem counterintuitive if you're thinking about motion on Earth. However, in the vacuum of space, slowing down the shuttle decreases its orbital energy, causing it to move to a lower orbit.

The crucial part of this operation is that it impacts the total mechanical energy of the shuttle. When a retrorocket is fired forward (in the direction of travel), it increases the magnitude of the negative total energy, leading to a decrease in orbit altitude. The firing must be carefully calculated to achieve the desired new orbit without destabilizing the overall path of the shuttle.

Total Mechanical Energy

Total mechanical energy in the context of orbital mechanics is the sum of kinetic and potential energies of an orbiting object. In a stable circular orbit, this energy remains constant, allowing the space shuttle to orbit around Earth indefinitely, assuming no external forces act upon it.

The mechanical energy is negative in a bound system, such as a shuttle orbiting the Earth, implying the shuttle is bound to the Earth by gravity. When a retrorocket is fired to slow down the shuttle, it increases the magnitude of this negative energy, meaning the shuttle now has less energy to stay far from Earth and moves to a lower orbit. This change directly correlates with the change in the shuttle's velocity, as both the mechanical energy and the orbit radius decrease. Each step in the solution calculates how the reduction in total energy by a retrorocket results in a new, lower circular orbit.