For “sufficiently large” objects moving “sufficiently fast” through a fluid, the

Question

For “sufficiently large” objects moving “sufficiently fast” through a fluid, the drag force on the object is proportional to the square of its velocity (quadratic drag). In this limit, we can ignore viscosity and argue this dependence from kinetic considerations.

a.) Suppose a sphere of radius R moves with speed v through a fluid with mass density . In a small time interval dt, what is the mass m of fluid that the sphere encounters?

b.) In this time interval, the ball pushes the mass m of fluid out of the way, by accelerating it to speed v (the speed of the ball itself). By Newton’s third law, this push exerts a backward force on the ball, which is the drag force.What is the magnitude of this force, in this model? How does the force scale with the cross-sectional area of the ball? (Note: you should get the correct scaling with area and velocity, but the coefficient in front requires a more sophisticated theory.)

Note: I'm having trouble relating the mass of the fluid to the velocity of the ball. Some conceptual tips will be helpful as well.

Explanation / Answer

(a) mass of fluid encountered by the sphere m = surface area of sphere * fluid density * speed of sphere * time interval

m = 4 R2 * * v * dt

(b) Drag force Fd = (1/2)CdAv2

Cd here is drag coefficient.

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