Two planets, Gamma and Delta, with masses \(m_G\) and \(m_D\) and radii \(R_G\) and \(R_D\) are initially at rest a large distance \(d\) apart from each other. Khan the Conqueror suddenly allows the two planets to approach under their mutual gravitational attraction until they collide. To determine the velocity at which Gamma and Delta collide you could (A) use conservation of energy alone. (B) use conservation of momentum alone. (C) use both conservation of energy and momentum. (D) The problem cannot be solved using energy and momentum conservation. (E) The problem cannot be solved unless you know how long it takes the two planets to collide.

Understanding the conservation of energy is crucial when analyzing physical systems, including the collision of celestial objects like in our exercise. This law states that energy cannot be created or destroyed in an isolated system. Within such a system, energy can change forms, such as from gravitational potential energy to kinetic energy, but the total amount remains constant.

In our problem involving two planets, we consider the entire system of these celestial bodies as isolated from external forces. Initially, when the planets are at rest and separated by distance d, the system's total energy is purely gravitational potential energy. As they move towards each other due to gravitational pull, this potential energy gets converted into kinetic energy.

The conversion can be modeled by the equation: $$E_{initial} = E_{final}$$ $$\frac{Gm_G m_D}{d} = \frac{1}{2} m_Gv_G^2 + \frac{1}{2} m_Dv_D^2$$Here, G represents the gravitational constant, m_G and m_D are the masses of planets Gamma and Delta, respectively, and v_G and v_D are their final velocities at the moment of collision. The formula links the initial gravitational potential energy with the final kinetic energies of both planets, exemplifying energy conservation in a closed system.

Another fundamental concept in physics is the conservation of momentum. Momentum, a quantity defined as the product of an object's mass and velocity, remains constant in an isolated system, absent of external forces. In the two-planet collision problem, since external forces are absent the total momentum of the system must stay constant.

Initially, both planets are at rest, implying that the initial momentum of the system is zero. As gravity causes them to move towards each other, their velocities increase, yet their combined momenta balance out to maintain zero net momentum.

The equation $$p_{initial} = p_{final}$$$$0 = m_G v_G + m_D v_D$$illustrates this balance, signifying that the momentum of planet Gamma (moving in one direction) and planet Delta (moving in the opposite direction) cancel each other out. Because momentum is a vector quantity, taking into account the direction is key, which is why the velocities carry positive and negative signs. The conservation of momentum helps us to link the speeds of the two planets and serves as a vital constraint in solving the collision problem.

Gravitational potential energy is the energy an object possesses due to its position in a gravitational field. In the context of our planetary collision, each planet has gravitational potential energy due to its separation from the other. The further apart they are, the higher the potential energy. This concept plays a significant role in our problem because it represents the 'stored' energy that gets converted to kinetic energy as the planets move closer.

The formula for gravitational potential energy between two masses is $$U = -\frac{Gm_1m_2}{r}$$ where U is the potential energy, G is the gravitational constant, m_1 and m_2 are the masses of the two objects, and r is the distance separating them. In our specific case of planets Gamma and Delta, initially their entire mechanical energy is potential in nature, which will diminish as they close in on each other due to their mutual attraction, converting it into kinetic energy.

Kinetic energy represents the energy of motion. An object acquires kinetic energy as it moves and the amount of this energy depends on both its mass and its speed. In physical formulas, the kinetic energy (KE) is expressed as $$KE = \frac{1}{2}mv^2$$ where m is the mass of the object and v is its velocity. In the exercise involving the colliding planets, as the mutual gravitational attraction draws the planets together, their velocities increase, resulting in a gain in kinetic energy.

Throughout the process, as the gravitational potential energy of the system decreases, the kinetic energy corresponding increases. This interplay between potential and kinetic energy highlights the transition of energy forms while honoring the principle of energy conservation. Moreover, the velocities of the planets at collision can ultimately be determined by analyzing how this kinetic energy is distributed between the two bodies, under the constraints imposed by both the conservation of energy and momentum.