Problem 805 - JEE Physics

Question

The maximum and minimum distance of a comet from the sun are \(8 \times 10^{12} \mathrm{~m}\) and \(1.6 \times 10^{12} \mathrm{~m} .\) If its velocity when nearest to the sun is \(60 \mathrm{~ms}^{-1}\), What will be its velocity in \(\mathrm{ms}^{-1}\) when it is farthest? (A) 6 (B) 12 (C) 60 (D) 112

Step-by-Step Solution

Answer

The velocity of the comet when it is farthest from the sun is \(12 \mathrm{ms}^{-1}\), which corresponds to answer choice (B).

Step 1: Write down the given values

We have the maximum distance (farthest) of the comet from the sun, \(r_{max} = 8\times10^{12} \mathrm{m}\), the minimum distance (nearest) of the comet from the sun, \(r_{min} = 1.6\times10^{12} \mathrm{m}\), and its velocity when nearest to the sun, \(v_{min} = 60 \mathrm{ms}^{-1}\). We need to find its velocity when farthest to the sun, \(v_{max}\).

Step 2: Use the conservation of angular momentum

Angular momentum is conserved in this scenario, which means the product of the comet's mass, velocity, and distance from the sun is constant. This can be written as: \(m r_{max} v_{max} = m r_{min} v_{min}\), where m is the mass of the comet. Note that the mass of the comet remains the same throughout its orbit.

Step 3: Solve for \(v_{max}\)

We need to find \(v_{max}\), so we can rewrite the previous equation as: \(v_{max} = \frac{m r_{min} v_{min}}{m r_{max}}\) Since the mass of the comet cancels out, we have: \(v_{max} = \frac{r_{min} v_{min}}{r_{max}}\) Now, plug in the given values to find \(v_{max}\): \(v_{max} = \frac{(1.6\times10^{12} \mathrm{m})(60 \mathrm{ms}^{-1})}{8\times10^{12} \mathrm{m}}\)

Step 4: Calculate the value of \(v_{max}\)

By doing the calculations, we get: \(v_{max} = \frac{96\times10^{12} \mathrm{m^2s^{-1}}}{8\times10^{12} \mathrm{m}} = 12 \mathrm{ms}^{-1}\) The velocity of the comet when it is farthest from the sun is \( 12 \mathrm{ms}^{-1}\), which corresponds to answer choice (B).

Key Concepts

Kepler's Laws

Kepler's laws of planetary motion are crucial in understanding how objects move in space, especially if they orbit around a significant mass like the sun. There are three main laws developed by Johannes Kepler:

• First Law (Law of Ellipses): This law tells us that the orbit of each planet is an ellipse, with the Sun at one of the two foci.
• Second Law (Law of Equal Areas): This law states that a line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time. This means planets move faster when closer to the Sun and slower when they are farther away.
• Third Law (Law of Harmonies): According to this law, the square of a planet's orbital period is proportional to the cube of the semi-major axis of its orbit. This links the time a planet takes to orbit the Sun with the size of its orbit.

Together, these laws explain why a comet's speed changes at different points in its orbit. Kepler's second law is particularly relevant here: as the comet gets closer to the Sun, it's pulled by gravity more strongly and moves faster. Conversely, it slows down as it moves further away, maintaining energy conservation.

Elliptical Orbits

Elliptical orbits play a pivotal role in understanding the motion of celestial bodies like comets and planets. An ellipse is an elongated circle, or an oval shape. In an elliptical orbit, a body moves around a central point in such a manner that the total distance from two fixed points (the foci) remains constant as it travels.
One of these foci is occupied by the sun in the case of planets or comets in our solar system. As a body moves through its orbit:

• Its speed varies depending on its distance from the main focus, which is the sun for our solar system's bodies.
• The closest point to the sun in an elliptical orbit is called the perihelion, while the farthest point is known as the aphelion.
• This variation in speed due to changing distances is elegantly explained by Kepler's second law.

The concept of elliptical orbits is fundamental as it dictates how velocity changes are predicted, highlighting how objects do not move uniformly in a gravitational field, but rather accelerate and decelerate as they move along their paths.

Cometary Motion

Cometary motion is a fascinating aspect of celestial dynamics, characterized by dramatic speed changes and long, stretched-out paths through the solar system. Comets often have elliptical orbits, which can be highly elongated, bringing them from the outer reaches of the solar system into close proximity with the Sun.
As they approach the Sun, comets increase in velocity due to the intense gravitational pull.

• This is where Kepler's second law comes into play, as these celestial bodies sweep out equal areas in equal time intervals, moving fastest when near the Sun.
• When they are at the perihelion, their minimum approach to the sun, they have their maximum velocity.
• Conversely, at aphelion, the point where they are the farthest from the Sun, they travel at their lowest speed.

These dynamic changes in speed and velocity are an application of the conservation of angular momentum. Because angular momentum is constant, any change in distance from the Sun necessitates a change in velocity. This ensures that as comets journey through different parts of their orbit, their motion adheres beautifully to the principles of celestial mechanics.