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

  • Thread starter Thread starter Benzoate
  • Start date Start date
  • Tags Tags
    Motion
Join the discussion
Registration is free. Start your own thread to ask a follow-up.
Sign up Log in
1 reply · 3K views
Benzoate
Messages
420
Reaction score
0

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.
 
Physics news on Phys.org

Discussion

Benzoate said:
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! :smile:

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

(otherwise, your method is fine :smile:)

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! :rolleyes:

"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(∞). :smile: