Saurabh Koranglekar
Last Activity: 5 Years ago
The question about the speed of a small body dropped into a tunnel that goes through the diameter of a planet is quite intriguing. To find the speed of the body when it reaches the center, we need to consider the gravitational force acting on it and how that force changes as it moves towards the center of the planet.
An Overview of Escape Velocity
Escape velocity, denoted as Ve, is the minimum speed needed for an object to break free from the gravitational pull of a celestial body without any additional propulsion. For a planet, this velocity depends on its mass and radius. However, when we drop a small body into a tunnel that goes through the planet, the situation is quite different from simply escaping the gravitational field.
Understanding Gravitational Force Inside the Planet
Inside a uniform sphere, the gravitational force acting on an object decreases linearly as it approaches the center. At the surface, the force is at its maximum, and at the center, it becomes zero. This behavior is derived from Newton's law of universal gravitation and the shell theorem.
Kinetic Energy and Motion
As the body falls into the tunnel, it converts gravitational potential energy into kinetic energy. The key points to consider are:
- At the surface, the body starts from rest, so its initial kinetic energy is zero.
- As the body falls, it accelerates due to gravity, gaining speed.
- When it reaches the center, all the potential energy has been converted into kinetic energy.
Energy Conservation Principle
To calculate the speed at the center, we can use the conservation of energy principle. The potential energy lost as the body falls is equal to the kinetic energy gained:
The gravitational potential energy at the surface is given by:
U = -G(Mm)/r, where M is the mass of the planet, m is the mass of the small body, r is the radius of the planet, and G is the gravitational constant.
As the body falls, its potential energy decreases, and when it reaches the center, it has maximum kinetic energy:
K.E. = (1/2)mv^2, where v is the speed of the body at the center.
Using the conservation of energy, we can set up the equation:
Potential Energy at surface = Kinetic Energy at center
Therefore, we can derive that the speed at the center of the planet is:
v = √(GM/r) = Ve/(√2)
Final Assessment of Options
Given the calculated speed at the center of the planet, we can now evaluate the answer choices:
- A. Zero - Incorrect, as the body will have gained speed.
- B. Ve/2 - Incorrect, as it does not match our calculation.
- C. Ve/(√2) - Correct, as derived from our energy conservation analysis.
- D. Ve - Incorrect, as it would exceed the escape velocity.
In summary, when the small body reaches the center of the planet, its speed will be Ve/(√2), making option C the correct answer. This elegantly illustrates how gravitational forces and energy transformations work in a uniform gravitational field. If you have any further questions or need clarification, feel free to ask!
