Question

A scientist working for a space agency has noticed that a Russian satellite of mass \(250 . \mathrm{kg}\) is on collision course with an American satellite of mass \(600 .\) kg orbiting at \(1000 . \mathrm{km}\) above the surface. Both satellites are moving in circular orbits but in opposite directions. If the two satellites collide and stick together, will they continue to orbit or crash to the Earth? Explain.

Short answer

Answer: To determine if the two satellites will continue to orbit or crash to Earth after colliding, we need to compare their final mechanical energy (ME_final) with their initial mechanical energy (ME_initial). If ME_final is greater than or equal to ME_initial, the combined satellite system will continue to orbit. If ME_final is less than ME_initial, the combined satellite system will lose altitude and eventually crash to Earth.

Step-by-step solution

Determine the initial velocities of the satellites

We need to find the initial velocities of both satellites. We know that they are on a circular orbit at a height of 1000 km above Earth's surface. The gravitational force provides the centripetal force in a circular orbit, which can be represented by: \(F_g = \dfrac{GMm}{r^2} = \dfrac{mv^2}{r}\) 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 center of the Earth to the satellite, and \(v\) is the velocity of the satellite. First, we need to find the value of \(r\). The Earth's radius is 6371 km. Thus, \(r = 6371 + 1000 = 7371 \ \text{km} = 7.371 \times 10^6 \ \text{m}\). For each satellite, we can now write the equation as: \(v = \sqrt{\dfrac{GM}{r}}\) Now, calculate the velocities for both satellites.

Calculate initial kinetic and potential energies

Now that we have the velocities of both satellites, we can calculate their initial kinetic and potential energies. The initial kinetic energy (KE) of each satellite is given by: \(KE = \dfrac{1}{2}mv^2\) And the initial gravitational potential energy (PE) of each satellite is given by: \(PE = -\dfrac{GMm}{r}\) Calculate the total initial mechanical energy (ME) for both satellites: \(ME_{initial} = KE_{Russian} + PE_{Russian} + KE_{American} + PE_{American}\)

Calculate the final mechanical energy and momentum

If the two satellites collide and stick together, we need to determine their final mechanical energy. First, calculate the final momentum of the system. Since both satellites are moving in opposite directions, momentum is conserved: \(p_{initial} = p_{final}\) \(m_{Russian}v_{Russian} - m_{American}v_{American} = p_{final}\) Solve for the final velocity, \(v_{final}\): \(v_{final} = \dfrac{p_{final}}{m_{Russian}+m_{American}}\) Now, calculate the final kinetic energy (KE): \(KE_{final} = \dfrac{1}{2} (m_{Russian}+m_{American})v_{final}^2\) And the final gravitational potential energy (PE) remains the same as in the initial state: \(PE_{final} = PE_{Russian} + PE_{American}\) Calculate the total final mechanical energy (ME): \(ME_{final} = KE_{final} + PE_{final}\)

Compare the initial and final mechanical energies

Now compare the initial and final mechanical energies. If the final mechanical energy (\(ME_{final}\)) is greater than or equal to the initial mechanical energy (\(ME_{initial}\)), then the combined satellite system will continue to orbit. If the final mechanical energy is less than the initial mechanical energy, then the combined satellite system will lose altitude and eventually crash to Earth. By comparing the values, we can answer whether the satellites will continue to orbit or crash to Earth after colliding and sticking together.

Key concepts

Circular Orbit Dynamics

Understanding circular orbit dynamics is crucial in predicting the movement of satellites and other celestial bodies. In circular orbit dynamics, an object moves in a circular path around a larger body due to the gravitational force acting as the centripetal force. This force is necessary to keep the object moving in a curved path rather than in a straight line, which is the natural tendency due to inertia.

For a satellite orbiting Earth, as in the provided exercise, the gravitational pull from Earth creates this required centripetal force. The equation \(F_g = \frac{GMm}{r^2} = \frac{mv^2}{r}\) represents the balance between the gravitational force (\(F_g\)) and the centripetal force needed for circular motion, where \(G\) is the gravitational constant, \(M\) the mass of Earth, \(m\) the mass of the satellite, \(r\) the radius of orbit from Earth's center, and \(v\) the satellite's velocity.

For a stable circular orbit, the satellite must maintain a velocity that ensures this balance. If the velocity is too high, the satellite would spiral away; if too low, it would be pulled in towards Earth. Understanding this equilibrium is critical in the context of satellite collisions and their aftermath.

Conservation of Momentum

The conservation of momentum is a fundamental principle in physics, stating that the total momentum of an isolated system remains constant in the absence of external forces. Momentum, a vector quantity, is the product of an object's mass and velocity, and it takes both magnitude and direction into account.

In a satellite collision, like the one described in the exercise, conservation of momentum plays a pivotal role. Although the satellites are moving in opposite directions, their momentums cancel each other when combined since momentum is conserved. This principle is expressed mathematically as \(m_{Russian}v_{Russian} - m_{American}v_{American} = p_{final}\). Here, we consider both the magnitudes and directions of the satellites' motion to calculate the final momentum (\(p_{final}\)).

The aftermath of such a collision is directly influenced by the conservation of momentum, determining the final velocity—and thus the fate—of the combined satellite mass post-collision. Whether the new entity maintains orbit or crashes will depend on whether conservation of momentum allows for enough velocity to sustain the required centripetal force.

Gravitational Potential Energy

Gravitational potential energy (PE) is the energy an object possesses by virtue of its position in a gravitational field. For satellites, this energy is negative, indicating that work must be done against the Earth's gravity to move the satellite away from Earth. The formula for the gravitational potential energy of a satellite is given by \(PE = -\frac{GMm}{r}\), where \(G\) is the gravitational constant, \(M\) is the mass of Earth, \(m\) is the mass of the satellite, and \(r\) is the distance from Earth's center to the satellite.

In the event of a satellite collision, the gravitational potential energy is an integral part of evaluating the system's behavior. As elucidated in the exercise, the gravitational potential energy before and after the collision remains unchanged because it solely depends on the satellites' mass and their distance from Earth, not on their velocities or directions of travel. This constancy implies that even after a collision, the combined satellite entity has the same initial potential to perform work against Earth’s gravity. However, the system’s ability to maintain orbit will be contingent upon whether the kinetic energy and gravitational potential energy collectively provide enough energy to avoid a decaying orbit.