Gauss S Green S And Stokes Theorems

To solve this problem, we need to understand the concept of terminal velocity in a viscous medium according to Stokes' Law. The terminal velocity for a spherical object of radius and density falling through a medium with viscosity is given by:

where:

• is the radius of the sphere
• is the density of the sphere
• is the density of the medium (negligible here)
• is the viscosity of the medium
• is the acceleration due to gravity

For the first ball with radius , the terminal velocity is .

For the second ball with the same mass but a radius of , let's consider the terminal velocity .

The mass of the sphere is proportional to the volume, which is proportional to . Hence, keeping the mass constant implies:

The density of the new sphere will be:

Now applying these changes to the terminal velocity formula for the second ball:

Substitute :

Thus:

Substituting the known relationship we find:

The terminal velocity for the second ball is .

Hence, the correct answer is .

To solve this problem, we will apply principles from fluid mechanics and the physics of motion under gravity.

Step 1: Calculate the terminal velocity of the ball in water

For a sphere moving through a fluid, the terminal velocity can be calculated using the formula for Stokes' Law:

• is the radius of the ball.
• is the density of the ball.
• is the density of water.
• is the acceleration due to gravity.
• is the coefficient of viscosity of water.

Substituting these values into the formula gives:

Calculating this yields approximations for simplification:

Step 2: Use energy conservation to find

The ball falls freely under gravity and so its potential energy converts to kinetic energy before entering the water:

Where is the mass of the ball, and is the terminal velocity.

Cancelling from both sides gives:

Hence,

Substitute the known values:

Conclusion: The height from which the ball must fall is approximately 2518 m. Therefore, the correct answer is 2518 m.