Question

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?

Short answer

Answer: 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).

Step-by-step solution

Understanding the terms perihelion and aphelion

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.

Relating kinetic energy, potential energy, and mechanical energy

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.

Locating the maximum kinetic energy and speed

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.

Locating the maximum gravitational potential energy

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.

Conclusion

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).

Key concepts

Kinetic Energy and Comet Orbits

When considering a comet orbiting the Sun, it's crucial to understand how kinetic energy behaves throughout the comet's path. Kinetic energy is the energy of motion, and it's given by the formula \[ K = \frac{1}{2}mv^2 \] where \( m \) is the mass of the comet and \( v \) is its speed.The comet travels faster at certain points in its elliptical orbit, which leads to changes in its kinetic energy. At perihelion, the point in the orbit where the comet is closest to the Sun, the comet has its maximum speed. This is due to the strong gravitational pull from the Sun. Because kinetic energy depends on speed, it is at its peak at perihelion. Meanwhile, at aphelion, where the comet is furthest from the Sun, the speed and thus kinetic energy are at their lowest. Key points about kinetic energy in comet orbits:

• Kinetic energy is highest at perihelion.
• Kinetic energy is determined by the comet's speed.
• Changes in kinetic energy correspond to changes in distance from the Sun.

Understanding Gravitational Potential Energy

Gravitational potential energy is another critical concept when studying the motion of comets. It represents the potential energy due to the gravitational attraction between the comet and the Sun. This energy is described by the equation:\[ U = -\frac{GMm}{r} \]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.The negative sign in the formula indicates that gravity is an attractive force and the energy becomes less negative (i.e., "more positive") as you move further from the Sun.At aphelion, the distance \( r \) is greatest, making the gravitational potential energy least negative or at its maximum value. Conversely, at perihelion, when the comet is closest to the Sun, the gravitational potential energy reaches its most negative value. Important aspects of gravitational potential energy:

• Gravitational potential energy is greatest at aphelion (least negative).
• It's most negative at perihelion.
• It varies inversely with distance from the Sun.

Perihelion and Aphelion Dynamics

Understanding the points of perihelion and aphelion in a comet's orbit is crucial for grasping the behavior of kinetic and gravitational potential energies.

Perihelion

At perihelion, the comet is nearest to the Sun. This proximity increases the gravitational pull, making the comet move fastest along its orbit. As a result, both speed and kinetic energy peak here. Meanwhile, because the comet is so close to the Sun, its gravitational potential energy reaches its most negative value.

Aphelion

In contrast, aphelion is where the comet is furthest from the Sun, reducing the gravitational pull. The comet slows down, which means both its speed and kinetic energy are at their minimum. However, due to the greater distance, gravitational potential energy is at its maximum (least negative). Key dynamics at perihelion and aphelion:

• Maximum speed and kinetic energy at perihelion.
• Minimum kinetic energy and maximum potential energy at aphelion.
• Changes in energy types illustrate the balance of forces in elliptical orbits.