A projectile is launched vertically upward from a planet of mass \(M\) and radius \(R ;\) its initial speed is twice the escape speed. Derive an expression for its speed as a function of the distance \(r\) from the planet's center.

The law of conservation of energy is a fundamental principle in physics stating that energy cannot be created or destroyed in an isolated system. Instead, energy is transformed from one form to another. For example, when a basketball is tossed into the air, its kinetic energy—the energy due to motion—is converted into gravitational potential energy—the energy due to its position—as it rises. At the peak of its trajectory, the motion halts momentarily, and all the kinetic energy has been converted into gravitational potential energy.

When dealing with projectile motion, as in the exercise provided, the law of conservation of energy allows us to track the energy transformations of the projectile. At launch, the projectile has maximum kinetic energy and minimal gravitational potential energy. As it ascends, kinetic energy decreases while gravitational potential energy increases. When applying this principle, we often use equations that relate the various forms of energy, enabling us to solve for unknown quantities such as speed at different positions during the projectile’s flight.

Gravitational potential energy (GPE) is the energy an object possesses due to its position in a gravitational field. The higher the object is above a reference point, such as the surface of a planet, the greater its gravitational potential energy. The formula to calculate GPE in the context of a planet and a projectile is given by \( U = -\frac{GMm}{r} \) where \(U\) is the gravitational potential energy, \(G\) is the gravitational constant, \(M\) is the mass of the planet, \(m\) is the mass of the projectile, and \(r\) is the distance from the center of the planet to the projectile.

In the exercise, we use the concept of gravitational potential energy to help determine the speed of the projectile at different distances from the planet's center. GPE plays a key role in the law of conservation of energy, as it's one form of energy that is converted into kinetic energy (and vice versa) as the projectile moves.

Escape velocity is the minimum speed needed for an object to 'break free' from the gravitational attraction of a celestial body, without further propulsion. This velocity depends on the mass and radius of the celestial body from which it is escaping. The formula to calculate escape velocity is: \( v_e = \sqrt{\frac{2GM}{R}} \) where \( v_e\) is the escape velocity, \(G\) is the gravitational constant, \(M\) is the mass of the celestial body, and \(R\) is its radius.

In this exercise, the projectile is launched with an initial speed twice that of the escape velocity, allowing us to determine its initial kinetic energy. The concept of escape velocity enhances our understanding of the projectile's behavior when it travels upward against the force of gravity. As the projectile rises, it slows down due to the increase in gravitational potential energy, indicative of the energy required to potentially escape the planet's gravitational pull. Even though the projectile in this scenario does not aim to escape but to reach a certain altitude, escape velocity still serves as a crucial benchmark for its initial kinetic energy.