For suciently large objects moving suciently fast through a

For \\suciently large\" objects moving \\suciently fast\" through a uid, 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) (2 points) Suppose a sphere of radius R moves with speed v through a uid with mass density . In a small time interval dt , what is the mass m of uid that the sphere encounters?

(b) (2 points) In this time interval, the ball pushes the mass m of uid 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 coecient in front requires a more sophisticated theory.)

Solution

a. mass= volume x density = area x length x density= 2 x pi x R^2 x vdt x density

b. force= mass x acceleration = mass x v/dt = 2 x pi x R^2 x vdt x density x v/dt = 2 x pi x R^2 x v^2 x density