Biology The Escherichia coli (E. coli ) bacterium propels...

Biology The Escherichia coli $(E$. coli $)$ bacterium propels itself through water by means of long, thin structures called flagella. If the force exerted by the flagella doubles, the velocity of the bacterium A. doubles. B. decreases by half. A 1 -kg wood ball and a 5 -kg lead ball have identical sizes, shapes, and surface characteristics. They are dropped simultaneously from a tall tower. Air resistance is present. How do their accelerations compare? A. The 1 -kg wood ball has the larger acceleration, but it cannot be calculated without more information. B. The 5 -kg lead ball has the larger acceleration, but it cannot be calculated without more information. C. The accelerations are the same. D. The 5 -kg ball accelerates at five times the acceleration of the $1-\mathrm{kg}$ ball. E. The 1 -kg ball accelerates at five times the acceleration of the $5-\mathrm{kg}$ ball.

Biology The Escherichia coli $(E$. coli $)$ bacterium propels itself through water by means of long, thin structures called flagella. If the force exerted by the flagella doubles, the velocity of the bacterium
A. doubles.
B. decreases by half.
A 1 -kg wood ball and a 5 -kg lead ball have identical sizes, shapes, and surface characteristics. They are dropped simultaneously from a tall tower. Air resistance is present. How do their accelerations compare?
A. The 1 -kg wood ball has the larger acceleration, but it cannot be calculated without more information.
B. The 5 -kg lead ball has the larger acceleration, but it cannot be calculated without more information.
C. The accelerations are the same.
D. The 5 -kg ball accelerates at five times the acceleration of the $1-\mathrm{kg}$ ball.
E. The 1 -kg ball accelerates at five times the acceleration of the $5-\mathrm{kg}$ ball.

Key Concepts

Low Reynolds Number Hydrodynamics

At microscopic scales, such as those encountered by bacteria in water, the Reynolds number is very low, meaning that viscous forces dominate inertial forces. In such an environment, motion is heavily resisted by the fluid's viscosity, and the linear relationship between the applied force and the resulting velocity is a direct consequence of Stokes' drag law.

Force-Velocity Relationship in Viscous Fluids

In low Reynolds number regimes, as described by Stokes' law, the velocity of an object moving through a fluid is proportional to the force applied when the drag force is linearly dependent on speed. Thus, if the applied force is doubled, and all other factors remain constant, the velocity is expected to double as well.

Newton's Second Law of Motion

Newton's second law, which states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass, is fundamental in analyzing both propulsion in fluids and free-fall motion. It provides the basis for understanding how forces affect motion regardless of the system's context.

Air Resistance and Free-Fall Acceleration

When objects fall through air, drag forces oppose gravity. For objects with identical sizes, shapes, and surface properties, the drag force relative to their weight will be similar. As a result, regardless of their mass differences, these objects tend to accelerate at nearly the same rate under gravity because the net force per unit mass remains comparable.

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