The Diving Gannet A gannet is one of the most effective diving birds. It hovers

Question

The Diving Gannet

A gannet is one of the most effective diving birds. It hovers about 34 m above the surface of the ocean and when it sees a school of fish under the surface it dives straight down, pulling in its wings just before it hits. Before it hits the water it is speeding up because gravity is essentially the only force acting on it. (It has a highly aerodynamic shape that reduces the air drag significantly.) To get some sense of what is happening to the gannet as it dives, we will model the gannet as a small dense sphere.

(a)

Assuming that the gannet falls only under the influence of gravity, calculate the speed v0 that it will be going when it hits the water. Before it hits the water it speeds up with an acceleration of g = 10 m/s2.

v0 = ________m/s

Explain your reasoning.

(b)

Inside the water the gannet is also acted on by a constant buoyant force from the water pushing it up. The buoyant force is independent of velocity and for simplicity, suppose it exactly cancels the gannet's weight. Moving underwater, it also feels a resistive (viscous) force proportional to its speed. Still modeling the gannet as a small sphere, if it enters the water at a speed of v0, which of the following graphs would describe how its velocity changes with time?

A B    C D E

Explain your choice.

(c)

For a small steel sphere, the weight is significantly greater than the buoyant force since a steel sphere at rest underwater will begin to sink. Which graph would you choose for the velocity of a steel sphere entering the water with a speed v0?

A B C D E

Briefly explain why you did or did not change your answer from the one you gave for (b).

(d)

If the viscous force on the steel sphere is

Fvwater sphere = bv,

the mass of the sphere is m and the gravitational field is g, find an expression for the terminal velocity of the sphere in terms of m, g, and b. (Enter the magnitude. Ignore the buoyant force, which is not very significant for a steel sphere.)

vT = ________

Show your reasoning.

Explanation / Answer

a) uisng vf^2 - vi^2 = 2ad

v0^2 - 0^2 = 2(10)(34)

in vertical it travels 34 m under the influence of gravity.

v0 = sqrt(680) = 26.08 m/s

b) suppose viscous force is -bv.

then,

Net force,

Fnet = mg - Fb - bv

mg = Fb (Given)

so Fnet = -bv

m dv/dt = - bv

(m / bv)dv = - dt

m ln(bv) / b = -t v is from v0 to v

ln (bv) - ln(bv0) = -bt/m

ln(v/v0) = - bt/m

v = v0e^(-bt/m)

so graph will be (e)

Ans. E

C) mg > Fb

Fnet = mg - Fb - bv

here mg - Fb = F0 > 0

Fnet = F0 - bv

so after sometime it will attain terminal speed and will move with constant velocity.

ANs ( D)

d) F = mg - bv

at terminal velocity , Fnet = 0

so mg - bv = 0

v = mg / b    (it is constant velocity)

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