"In theory, theory and practice are the same. In practice, they are not."
..."Albert Einstein (1879~1955, Mathematical physicist, Nobel Prize 1921-Physics)"

Q: Suppose that a body moves in a medium which offers drag to its motion, the drag force F being proportional to the velocity v of the body, raised to some power b, that is, F = av^{b} If the body is imparted an initial velocity, is it possible for it to cover an infinite distance?

Let us rephrase this question a bit. You have to figure out whether there is any value of b possible for which if the body is given an initial non-zero velocity, it never stops.

The question might seem strange. After all, if there is drag on the body, its velocity should be reduced continuously and finally become zero in some finite amount of time, so it should eventually stop. Well, there is no doubt about the truth of the statement “Its velocity should be reduced continuously”. But the second part, “The velocity finally becomes zero in some finite amount of time”, might not be very true. This is what you have to think about.

« Last Edit: May 03, 2009, 10:02:53 pm by dgktiit »

Reply #1 on: May 03, 2009, 10:04:08 pm » posted from:Taipei,T\'ai-pei,Taiwan

It depends on if there is an external force acting on the object or not. If there is no other force, the drag force will reduce the velocity to zero- it is stopped! If there is a constant force, the drag force will only make the object reach a terminal valocity. i.e. F_{total}= F_{ext} -av^{b} then av^{b}=F_{ext} so terminal velocity v_{t}=e ^{log(Fext/a)/b}

no there is no external force...there is only drag force and the body is just imparted a velocity initially. since drag force is velocity dependent ...also velocity will reduce continuosly...so will dragging force...

Q: Suppose that a body moves in a medium which offers drag to its motion, the drag force F being proportional to the velocity v of the body, raised to some power b, that is, F = av^{b} If the body is imparted an initial velocity, is it possible for it to cover an infinite distance?

Let us rephrase this question a bit. You have to figure out whether there is any value of b possible for which if the body is given an initial non-zero velocity, it never stops.

The question might seem strange. After all, if there is drag on the body, its velocity should be reduced continuously and finally become zero in some finite amount of time, so it should eventually stop. Well, there is no doubt about the truth of the statement “Its velocity should be reduced continuously”. But the second part, “The velocity finally becomes zero in some finite amount of time”, might not be very true. This is what you have to think about.

sounds like a question for his/her undergraduate course?

if let a = 0 Fdrag= av^{b} =0

so Free Body Diagram apply Newton 2 law:

F = m*a - Fdrag = m*a 0 = m*a 0 = a It is a mathematical answer to achieve velocity to never decay by letting constant a =0, i doubt it can be achieve in real life Earth bound conditions

« Last Edit: May 04, 2009, 08:34:49 am by lookang »

Q: Suppose that a body moves in a medium which offers drag to its motion, the drag force F being proportional to the velocity v of the body, raised to some power b, that is, F = av^{b} If the body is imparted an initial velocity, is it possible for it to cover an infinite distance?

Let us rephrase this question a bit. You have to figure out whether there is any value of b possible for which if the body is given an initial non-zero velocity, it never stops.

The question might seem strange. After all, if there is drag on the body, its velocity should be reduced continuously and finally become zero in some finite amount of time, so it should eventually stop. Well, there is no doubt about the truth of the statement “Its velocity should be reduced continuously”. But the second part, “The velocity finally becomes zero in some finite amount of time”, might not be very true. This is what you have to think about.

sounds like a question for his/her undergraduate course?

if let a = 0 Fdrag= av^{b} =0

so Free Body Diagram apply Newton 2 law:

F = m*a - Fdrag = m*a 0 = m*a 0 = a It is a mathematical answer to achieve velocity to never decay by letting constant a =0, i doubt it can be achieve in real life Earth bound conditions

Hey! The drag is non-zero. That’s the whole point of this problem.

Reply #8 on: May 26, 2009, 04:33:40 pm » posted from:Taipei,T\'ai-pei,Taiwan

Since the drag is a*v^{b}, it means that F=-a*v^{b}, here assume a>0. assume b>1; F=m dv/dt; dv/dt= -a*v^{b}/m; v^{-b} dv= -(a/m) dt do integration for both side (1/(-b+1))v^{-b+1}-(1/(-b+1))v_{o}^{-b+1}=-at/m

v^{-b+1}= v_{o}^{-b+1}+a*(b-1)*t/m or v^{b-1}= 1/ (1/ v^{b-1}+a*(b-1)*t/m);

The denominator will become infinity when t approach to infinity, so v become zero.

"In theory, theory and practice are the same. In practice, they are not."
..."Albert Einstein (1879~1955, Mathematical physicist, Nobel Prize 1921-Physics)"