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Physics, Chemistry and Math!
Physics: Problem of the day – 11
This problem is again a challenge for you.
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 = avb
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.
May 4, 2009 - 10:30 AM
if let a = 0
Fdrag= av^b =0
so apply Newton 2nd 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 achieved in real life Earth bound conditions.
please temme if ,votever i did was right or …something stupidity.. ????
May 4, 2009 - 10:38 AM
if let a = 0
Fdrag= av^b =0
so apply Newton 2nd 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 achieved in real life Earth bound conditions.
please temme if ,votever i did was right or …something stupidity.. ????
May 4, 2009 - 3:26 PM
Hey! The drag is non-zero. That’s the whole point of this problem.
This means that a is not zero.
May 4, 2009 - 4:21 PM
ohhk sr.. please dont put up the sol soon.. i mean i wnt 2 giv it another chance…
May 5, 2009 - 1:07 PM
I can think of a related phenomena, but not an exact value for b… like suppose we project a body from the surface of the earth with velocity ≥11.2 km/s, as the body goes higher, its velocity will decrease, and so will the force on it (gravitation due to earth)… so the retarding force is proportional to some power of velocity…but theoritically it will travel infinite distance. correct me if i’m wrong..
May 5, 2009 - 1:28 PM
Hey Dylan, the force dependence on velocity in the case you’ve suggested will be slightly more complicated than the relation mentioned here.
In this problem, we can find an exact set of values of b for which the body will be able to cover an infinite distance.
May 22, 2009 - 7:18 PM
is the ans. b=2
May 23, 2009 - 11:48 AM
Yeah. Good work! How did you arrive at the answer?
May 26, 2009 - 7:54 PM
yes
May 26, 2009 - 7:55 PM
because drag force depends on velocity when velocity of particle become zero then drag force is zero hence finally block will be at rest.
May 27, 2009 - 2:00 PM
Since the drag is a*vb, it means that F=-a*vb, here assume a>0.
assume b>1;
F=m dv/dt;
dv/dt= -a*vb/m;
v-b dv= -(a/m) dt
do integration for both side
(1/(-b+1))v-b+1-(1/(-b+1))vo-b+1=-at/m
v-b+1= vo-b+1+a*(b-1)*t/m
or
vb-1= 1/ (1/ vb-1+a*(b-1)*t/m);
The denominator will become infinity when t approach to infinity, so v become zero.
May 27, 2009 - 2:10 PM
Hi Nikhil, the answer is that for b >=2, the particle can travel an infinite distance.
May 27, 2009 - 2:11 PM
Since the drag is a*vb, it means that F=-a*vb, 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^(-b+1)=-at/m
v^(-b+1)= V^(-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.
May 27, 2009 - 2:11 PM
“Cnt tell u” –> read the comment above.
May 27, 2009 - 2:15 PM
hmm..i did not read that. .. but wats wrong in my solution .. ?
m not able to figure it out. .if u can then i will be grateful.
May 27, 2009 - 5:13 PM
if b=1, the integral will be ln x and not x^(n+1)/n+1
May 27, 2009 - 11:18 PM
i think u didnt read my assumption. .
i assumed that b>1. . .
May 28, 2009 - 10:53 AM
Hmm…you are right that v will approach 0. Does that prove that the particle will travel only a finite distance? Think of it this way. The velocity decreases with time and approaches zero, but nevertheless the distance keeps on increasing, no matter how slowly. So it may be possible that in a sufficiently large time interval ( –> infinite time), the particle travels an arbitrarily large distance ( –> infinite distance) even though v –> 0.
May 28, 2009 - 11:52 AM
exactly thats wat i wanted to say. .. so is the mathematical way ..the way i solved..correct .. ??
May 29, 2009 - 11:59 AM
wat happened sr.. ?????
plz reply.
May 29, 2009 - 2:10 PM
You never answered the actual question –> the actual question is, can the particle travel an infinite distance? You only proved that v will finally approach zero, which is true, but the original question was to find those values of b, if there exist any, for which the particle can travel an infinite distance.