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 \(10.0 \%\) (that is, increasing the magnitude of the negative total energy by \(10.0 \%\) ), 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

Solution: (a) The initial speed of the space shuttle can be calculated using the formula: \(v_{i} = \sqrt{r \cdot a_g}\) (b) The final speed of the space shuttle can be found using the formula: \(v_{f} = \sqrt{\frac{2K_{f}}{m}}\) Calculate the initial and final speeds by substituting the known values for \(r\), \(a_g\), and \(K_{f}\) into the respective formulas.

Step-by-step solution

Gravitational force and initial potential energy

We begin by calculating the gravitational force acting on the space shuttle, which is given by: \(F = G \frac{m M}{r^2}\) Where \(F\) is the gravitational force, \(G\) is the gravitational constant \(G=6.674 \times 10^{-11} \mathrm{N \cdot m^2/kg^2}\), \(m\) is the space shuttle's mass, \(M\) is Earth's mass \(M=5.97 \times 10^{24} \mathrm{kg}\), and \(r\) is the distance between the centers of the shuttle and Earth. The initial potential energy of the space shuttle can be given as: \(U = -G \frac{m M}{r}\)

Gravitational acceleration and initial kinetic energy

Next, we calculate the gravitational acceleration acting on the space shuttle: \(a_g = \frac{F}{m} = G \frac{M}{r^2}\) Now, use the centripetal acceleration formula to find \(v_{i}\), the initial speed of the space shuttle: \(a_c = \frac{v_{i}^2}{r}\) Since \(a_c = a_g\), we can find the initial velocity of the space shuttle: \(v_{i}^2 = r \cdot a_g \Rightarrow v_i = \sqrt{r \cdot a_g}\) Now we can calculate the initial kinetic energy of the space shuttle: \(K_i = \frac{1}{2} mv_i^2\)

Total initial energy and reduced energy

The total initial mechanical energy of the space shuttle is a sum of kinetic and potential energy: \(E_{i} = K_i + U\) Now we know that the retrorocket reduces the total energy by 10%, so the final total energy \(E_f\) can be expressed as: \(E_{f} = E_{i} - 0.1E_{i} = 0.9E_{i}\)

Final potential and kinetic energy

Just like with the initial energy, the final total mechanical energy is also the sum of potential and kinetic energy: \(E_{f} = K_f + U_f\) We know that the final orbital radius \(r'\) must be smaller than the initial radius \(r\). We can find the total final potential energy: \(U_{f} = -G\frac{mM}{r'}\) Now, we can use conservation of mechanical energy to find the final kinetic energy: \(K_{f} = E_{f} - U_{f}\)

Final velocity and solution

The final kinetic energy can be expressed as: \(K_{f} = \frac{1}{2} m v_{f}^2\) Since we know \(K_{f}\), we can solve for the final velocity, \(v_{f}\): \(v_{f}^2 = \frac{2K_{f}}{m} \Rightarrow v_{f} = \sqrt{\frac{2K_{f}}{m}}\) Now we can finally find the space shuttle's initial and final speed: (a) The initial speed can be calculated using the formula found in Step 2: \(v_{i} = \sqrt{r \cdot a_g}\) (b) The final speed can be found using the formula derived in the last step: \(v_{f} = \sqrt{\frac{2K_{f}}{m}}\) In both cases, substitute the known values and solve for \(v_{i}\) and \(v_{f}\) respectively to find the speed of the space shuttle before and after the retrorocket is fired.

Key concepts

Gravitational Force

When learning about how objects move in space, we first encounter the invisible tug that maintains their celestial dance: the gravitational force. This force is the attraction between any two masses. For instance, the Earth pulls on a space shuttle using gravity, an effect explained by Isaac Newton's law of universal gravitation. In the problem, we calculate it using the equation
\(F = G \frac{m M}{r^2}\)
This formula encapsulates the gravitational force that the Earth exerts on the space shuttle in orbit, where \(G\) is the gravitational constant, \(m\) signifies the space shuttle's mass, \(M\) represents Earth's mass, and \(r\) is the distance from the Earth's center to the shuttle. Understanding this force is crucial because it's what keeps the space shuttle — or any satellite — in orbit, as it competes against the object's velocity trying to send it hurtling away into space.

Orbital Mechanics

Orbital mechanics, or celestial mechanics, is the field of physics focused on the motions of celestial bodies under the influence of gravitational forces. It explains the principles behind the space shuttle's orbit prior to the retrorocket firing. The balance between the shuttle’s velocity and Earth's gravitational pull keeps it in a stable orbit. When a satellite or a shuttle is in orbit, the gravitational force provides the necessary centripetal force that leads to centripetal acceleration. If the shuttle's speed changes, its orbit will adjust accordingly to maintain this delicate balance. In our exercise, the shuttle transitions to a new orbit with a different radius after the retrorocket fires, changing its total mechanical energy and velocity, a principle crucial to orbital maneuvers in spaceflight.

Conservation of Mechanical Energy

The principle of the conservation of mechanical energy states that if no external work is done on a system, its total mechanical energy remains constant. However, mechanical energy can transform between its two forms: potential energy, associated with the position of an object in a gravitational field, and kinetic energy, associated with the motion of the object. In our scenario, the space shuttle's mechanical energy changes due to the external work done by the retrorocket.
In space, when an engine like a retrorocket applies a force, it alters the energy of the spacecraft. The initial potential and kinetic energy are reduced after firing. After finding the total initial energy \(E_{i}\), we calculate the energy reduced by the rocket's burn as 10%, which leads us to the final total energy \(E_{f}\). Thus, despite the mechanical energy conversion from kinetic to potential energy as the shuttle moves into a lower orbit, the conserved quantity is the total mechanical energy, minus the work done by the retrorocket.

Centripetal Acceleration

Centripetal acceleration is what keeps a space shuttle in a circular orbit. As the shuttle travels around Earth, it constantly accelerates toward Earth's center, even if its speed remains constant, because its velocity's direction is continuously changing. Centripetal acceleration refers to this continuous change in velocity direction. For circular orbits, centripetal acceleration is provided purely by the gravitational pull of the Earth and can be computed using the formula
\(a_c = \frac{v^2}{r}\)
where \(v\) is the orbital speed and \(r\) the orbit radius. In our exercise, by equating gravitational acceleration \(a_g\) to centripetal acceleration \(a_c\), we can derive the initial velocity of the space shuttle. Later, after energy is removed by the retrorocket, we can also determine the new velocity, since the centripetal acceleration must adjust to the new radius of the shuttle's orbit, exemplifying how a change in energy translates into a change in the shuttle's orbit speed and its centripetal acceleration.