Final answer:
While the cross-sectional area of a ball with twice the radius is four times greater, the terminal velocity is proportional to the square root of the radius. Therefore, none of the provided options accurately represent the ratio of their terminal velocities.
Step-by-step explanation:
When comparing the terminal velocities of two balls of different sizes, we need to consider several factors, but one of the most important is the drag force experienced by the balls as they fall through a fluid (like air). The drag force can be described by the equation:
Fd = ½CdρAv2
where Fd is the drag force, Cd is the drag coefficient, ρ is the density of the fluid, A is the cross-sectional area, and v is the velocity of the object.
For spherical objects like balls, the cross-sectional area is proportional to the square of the radius (A = πr2). Since the radius of ball A is twice that of ball B, the cross-sectional area of A is four times that of B. However, the terminal velocity is reached when the drag force equals the gravitational force, and since the gravitational force is proportional to the mass (which, for a sphere of constant density, is proportional to the cube of the radius), we must also consider this factor.
The terminal velocity, vt, can be derived from the balance of forces and is proportional to the square root of the radius (vt √ r), meaning that if the radius of ball A is twice that of ball B, the terminal velocity of A will be √2 times that of B. Therefore, none of the provided options (A: 1:1, B: 1:2, C: 2:1, D: 4:1) correctly represents the relationship between the terminal velocities of the two balls.