A comet orbiting the Sun moves in an elliptical orbit. Where is its kinetic energy, and therefore its speed, at a maximum, at perihelion or aphelion? Where is its gravitational potential energy at a maximum?

In an elliptical orbit, the perihelion is the smallest distance between the orbiting object (in this case, the comet) and the center of the orbit (the Sun). Aphelion, on the other hand, is the greatest distance between the object and the center of the orbit. In other words, the comet is closest to the Sun at perihelion and farthest away at aphelion.

As the comet orbits the Sun, its mechanical energy (the sum of its kinetic energy and gravitational potential energy) remains constant. This is due to the conservation of mechanical energy. Mathematically, we can write this as: E = K + U where E is the total mechanical energy, K is kinetic energy, and U is gravitational potential energy. For the kinetic energy, K = (1/2)mv^2, where m is the mass of the comet and v is its speed at any point in the orbit. For gravitational potential energy, U = -(GMm)/r, where G is the gravitational constant, M is the mass of the Sun, m is the mass of the comet, and r is the distance between the comet and the Sun.

As the distance between the comet and the Sun (r) changes between perihelion and aphelion, the gravitational potential energy (U) will also change. Since the mechanical energy (E) must remain constant, any change in U must be balanced by an equal and opposite change in the kinetic energy (K). When the comet is closest to the Sun (at perihelion), the value of r is smallest. The gravitational potential energy, U = -(GMm)/r, is inversely proportional to r, so U will be most negative (or least) at perihelion. Since the mechanical energy (E) must remain constant, the kinetic energy (K) must be at a maximum to balance the minimum potential energy. Therefore, the speed of the comet will also be at a maximum at perihelion.

When the comet is farthest from the Sun (at aphelion), the distance r is greatest. Since gravitational potential energy is inversely proportional to r, the potential energy (U) will be at a maximum (least negative) at this point. It's important to note that since the comet is farther from the Sun, the magnitude of the gravitational potential energy will still be negative but less so than at perihelion.

From our analysis, we find that the kinetic energy and speed of the comet are at a maximum at its perihelion (when it is closest to the Sun) and the gravitational potential energy is at a maximum (least negative) at its aphelion (when it is farthest from the Sun).