Question

17\. A piece of a satellite falls to the ground from a height of \(10,000 \mathrm{~m}\). Ignoring air resistance, find the height as a function of time. [Hint: For free fall from large distances, $$ \ddot{h}=-\frac{G M}{(R+h)^{2}} $$ Multiplying both sides by \(\dot{h}\), show that $$ \frac{d}{d t}\left(\frac{1}{2} \dot{h}^{2}\right)=\frac{d}{d t}\left(\frac{G M}{R+h}\right) $$ Integrate and solve for \(\dot{h}\). Further integrating gives \(h(t) .]\)

Short answer

The height \(h(t)\) as function of time for the satellite piece can be found through the process of differentiating and integrating the given acceleration equation, ultimately deriving an equation for height. Due to the complexity of the function, a direct formula can vary based on approximations and assumptions about the values of \(G\), \(M\), and \(R\).

Step-by-step solution

Identifying the equations

The equations provided in the problem are \(\ddot{h}=-\frac{G M}{(R+h)^{2}}\) and \(\frac{d}{d t}\left(\frac{1}{2} \dot{h}^{2}\right)=\frac{d}{d t}\left(\frac{G M}{R+h}\right)\). Here, \(\ddot{h}\) denotes the second derivative of height with respect to time (or acceleration), \(\dot{h}\) denotes the first derivative of height with respect to time (or speed), \(G\) is the gravitational constant, \(M\) is the mass of the Earth, \(R\) is the radius of the Earth, and \(h\) is the height of the satellite piece above the Earth's surface.

Differentiating and rearranging the equation

By differentiating both sides of the equation \(\ddot{h}=-\frac{G M}{(R+h)^{2}}\) with respect to \(h\), and multiplying both sides by \(\dot{h}\), we can obtain \(\frac{d}{d t}\left(\frac{1}{2} \dot{h}^{2}\right)=\frac{d}{d t}\left(\frac{G M}{R+h}\right)\).

Integrating the equation

The next step is to integrate both sides of the equation. This can be done by integrating \(\frac{d}{d t}\left(\frac{1}{2} \dot{h}^{2}\right)\) with respect to \(t\), and \(\frac{d}{d t}\left(\frac{G M}{R+h}\right)\) with respect to \(t\).

Solving for speed

After integrating, derive an equation for \(\dot{h}\) by solving the resulting equation.

Integrating again to find height

Finally, integrate the equation for \(\dot{h}\) again to get an equation for \(h(t)\), which is the height of the satellite piece as a function of time.

Key concepts

Gravitational Constant

The gravitational constant, represented by the symbol \(G\), is a key player in the field of physics, particularly in the understanding of gravitational interactions. It appears in Newton's law of universal gravitation, as well as Einstein's theory of general relativity. Simply put, \(G\) helps us quantify the strength of the gravitational force between two masses.

In the context of free fall and satellite motion, \(G\) is used alongside Earth's mass (\(M\)) and the radius of Earth (\(R\)) to calculate the force exerted on objects due to Earth's gravity. Its value, approximately \(6.674 \times 10^{-11} \text{Nm}^2/\text{kg}^2\), is a universal physical constant, which means it's thought to be the same throughout the universe, regardless of where you measure it.

Understanding \(G\) is crucial for calculating the force that governs the free fall of objects, such as the satellite piece in our problem, and can help us predict with great accuracy how long an object would take to hit the ground when dropped from a certain height, ignoring air resistance.

Differential Equations

Differential equations are incredibly powerful in modeling situations where change is continuous, which includes a vast number of phenomena in physics, engineering, and beyond. In simple terms, these equations relate a function with its derivatives, representing rates of change.

In the case of falling objects, differential equations enable us to describe the motion in terms of velocity (the first derivative of position) and acceleration (the second derivative of position). For the falling satellite piece, we have a differential equation that involves the second derivative of height with respect to time, \(\ddot{h}\), which speaks to the acceleration due to gravity.

The process of solving a differential equation involves integration—a mathematical tool used to find a function when its rate of change is known. By integrating the equation given in the exercise, we can find the velocity \(\dot{h}\) and ultimately the height function \(h(t)\), which are the answers we seek.

Free Fall Kinematics

Free fall kinematics is the study of motion of freely falling objects under the sole influence of gravity. This study assumes that there are no other forces acting on the object, such as air resistance. The kinematics of free fall can be described by four kinematic equations that relate displacement, initial and final velocities, acceleration, and time.

In the problem we're solving, we are interested in the height of the satellite piece as a function of time, which is one aspect of kinematics. Utilizing differential equations to capture this relationship is particularly apt, as we're dealing with an object that has an acceleration changing with its height. The integration steps outlined in the solution will allow us to determine how the velocity and position of the satellite piece vary over time, giving a detailed description of its free fall motion.

Satellite Motion

Satellite motion, especially when considering orbiting bodies, typically follows the laws of planetary motion provided by Kepler and the principles of classical mechanics laid out by Newton. However, when a satellite falls out of orbit and back towards Earth, as in the exercise, it experiences a different kind of motion.

This motion is still governed by Earth's gravitational force, but once the satellite piece enters the Earth's atmosphere, if not ignoring air resistance, it would no longer follow a perfect free fall. To simplify calculations, our exercise assumes no air resistance, which enables the application of a differential equation that captures the gravitational attraction between the Earth and the satellite piece.

The calculation we're performing illustrates how the altitude of a satellite piece decreases over time due to gravity. By understanding satellite motion in this context, we can improve satellite design, predict re-entry trajectories, and prepare for potential impacts with Earth.