Problem 73 - University Physics with Modern Physics

Question

Comets travel around the sun in elliptical orbits with large eccentricities. If a comet has speed \(2.0 \times 10^{4} \mathrm{m} / \mathrm{s}\) when at a distance of \(2.5 \times 10^{11} \mathrm{m}\) from the center of the sun, what is its speed when at a distance of \(5.0 \times 10^{10} \mathrm{m} ?\)

Step-by-Step Solution

Answer

The speed of the comet is approximately \( 3.16 \times 10^4 \mathrm{m/s} \) at a distance of \( 5.0 \times 10^{10} \mathrm{m} \).

Step 1: Understand the Conservation of Energy

The total mechanical energy of the comet, when considering its orbit, is conserved. This means that the sum of its kinetic and potential energy at one point in its orbit is equal to the sum at any other point. Mathematically, this can be expressed as \( \frac{1}{2}mv_1^2 - \frac{GMm}{r_1} = \frac{1}{2}mv_2^2 - \frac{GMm}{r_2} \), where \(v_1\) and \(v_2\) are speeds at distances \(r_1\) and \(r_2\), respectively, and \(G\) is the gravitational constant, \(M\) the mass of the sun.

Step 2: Simplify the Equation

The mass of the comet \(m\) appears in every term, so it can be canceled out. The equation now becomes: \( \frac{1}{2}v_1^2 - \frac{GM}{r_1} = \frac{1}{2}v_2^2 - \frac{GM}{r_2} \). Rearranging gives us \( \frac{1}{2}v_1^2 + \frac{GM}{r_2} = \frac{1}{2}v_2^2 + \frac{GM}{r_1} \).

Step 3: Isolate the Variable for Final Speed

Solve for \(v_2^2\) by rearranging the equation: \( v_2^2 = v_1^2 + 2GM \left( \frac{1}{r_2} - \frac{1}{r_1} \right) \). This equation allows for the calculation of the comet's speed at the second position using initial known values.

Step 4: Substitute the Known Values

Insert the known values: \(v_1 = 2.0 \times 10^4 \mathrm{m/s}\), \(r_1 = 2.5 \times 10^{11} \mathrm{m}\), \(r_2 = 5.0 \times 10^{10} \mathrm{m}\), and \(G = 6.674 \times 10^{-11} \mathrm{m^3/kg \, s^2}\), \(M = 1.989 \times 10^{30} \mathrm{kg}\) (mass of the Sun):

Key Concepts

Conservation of Energy

In the context of comets orbiting the sun, the principle of conservation of energy plays a crucial role. It states that the total mechanical energy of an object—like a comet—remains constant if only conservative forces, such as gravity, act upon it. This means that the sum of kinetic energy (energy due to motion) and potential energy (energy due to position) remains unchanged as the comet moves along its orbit.

To calculate the speed of a comet at different points in its orbit, we use the equation:

• \( \frac{1}{2}mv_1^2 - \frac{GMm}{r_1} = \frac{1}{2}mv_2^2 - \frac{GMm}{r_2} \)

In this equation,

• \(m\) is the mass of the comet,
• \(v_1\) and \(v_2\) are the comet's speeds at distances \(r_1\) and \(r_2\) from the center of the sun, and
• \(G\) is the gravitational constant and \(M\) is the mass of the sun.

In calculations involving energy conservation, observing that the mass \(m\) of the comet cancels out aids in simplifying the solution. This highlights that the speeds are solely dependent on the distances and the massive gravitational pull of the sun.

Elliptical Orbits

Elliptical orbits are a characteristic feature of planetary motion, describing the paths followed by comets around the sun. Unlike circular orbits, elliptical orbits are elongated, with the sun located at one of the foci of the ellipse. This causes variations in the speed of the comet as it travels through different regions of its orbit.

One important aspect of elliptical orbits is Kepler's First Law, which states that "The orbit of a planet (or comet) is an ellipse with the Sun at one of the two foci." This law helps explain why comets move faster when they are closer to the sun (at the perihelion) and slower when they are farther away (at the aphelion).

Understanding these orbits helps determine how gravitational forces and energy dynamics influence the motion of celestial bodies. It shows why the speed of a comet changes as it moves along its elliptical path and helps in predicting its position at given times.

Gravitational Forces

Gravitational forces are the invisible agents that govern the movement of celestial bodies such as comets in the universe. According to Newton's Law of Universal Gravitation, every two objects with mass attract each other with a force that is proportional to the product of their masses and inversely proportional to the square of the distance between their centers. The formula is expressed as:

• \( F = \frac{G \, M \, m}{r^2} \)

In this formula:

• \(F\) is the magnitude of the gravitational force,
• \(G\) is the gravitational constant \( (6.674 \times 10^{-11} \, \text{m}^3/\text{kg} \, \text{s}^2)\),
• \(M\) and \(m\) are the masses of the two objects (e.g., the Sun and a comet), and
• \(r\) is the distance between the centers of the two masses.

For comets, these gravitational forces dictate their intricate dance around the sun. The interplay of these forces and the inertia of the comet results in the observed elliptical orbits. Understanding gravitational forces allows scientists to predict the trajectory and speed changes of comets as they move within the gravitational fields of larger celestial bodies.