Analysis of Motion in a Medium with Resistance Force: Case I and II

[Benzoate]

Homework Statement

A body of mass is projected with speed and moves under uniform gravity in a medium that exerts a resistance force of magnitude (i) mk*abs(v) or (ii) mK*(abs(v))^2 , where k and K are the positive constants and v is the velocity of the body. Gravity can be ignored. Determine the subsequent motion in each case . Verify motion the motion is bounded in case i , but not in case ii

Homework Equations

F=D+L

The Attempt at a Solution

for case i m*dv/dt= m*k*abs(v)

applying separation of variables, I get

dv/abs(v)=kdt ==> ln(abs(v))=kt ===> v=Ce^-kt or v=Ce^kt , C being a constant and v depending on whether or not is positive or negative.

for case two, my physical system is the quadratic resistance

dv/dt=mk*(abs(v))^2

applying once again the separation of variables method I get:

-1/K*1/v=t ==> v=-1/kt+C

I don't understand how to verify that the body is bounded. I know that a body is bounded if it cannot overcome its gravitational potential energy. I don't understand how I can possibly ignore gravity , unless the body is completely immersed in a vacuum and that cannot be possible if a fluid force is exerting a force on the object.

 

[tiny-tim]

A body of mass is projected with speed and moves under uniform gravity in a medium that exerts a resistance force of magnitude (i) mk*abs(v) or (ii) mK*(abs(v))^2 , where k and K are the positive constants and v is the velocity of the body. Gravity can be ignored. Determine the subsequent motion in each case . Verify motion the motion is bounded in case i , but not in case ii
…
for case i m*dv/dt= m*k*abs(v)

applying separation of variables, I get

dv/abs(v)=kdt ==> ln(abs(v))=kt ===> v=Ce^-kt or v=Ce^kt , C being a constant and v depending on whether or not is positive or negative.

for case two, my physical system is the quadratic resistance

dv/dt=mk*(abs(v))^2

applying once again the separation of variables method I get:

-1/K*1/v=t ==> v=-1/kt+C

I don't understand how to verify that the body is bounded. I know that a body is bounded if it cannot overcome its gravitational potential energy. I don't understand how I can possibly ignore gravity , unless the body is completely immersed in a vacuum and that cannot be possible if a fluid force is exerting a force on the object.

Hi Benzoate!

First … big mistake … resistance acceleration is negative, isn't it?

(otherwise, your method is fine )

As to gravity, I don't understand why they tell you that the gravity is uniform, and then tell you to ignore it! Well, ignore it anyway!

"Unbounded" means that the body goes infinitely far (in other words, x(∞) = ∞). To check that, put v = dx/dt, and integrate again to find x(∞).