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Consider an aeroplane of mass M which is flying horizontally ( neglect gravity) with velocity v and dust, which has velocity u in the opposite direction to the plane get stuck to the plane. We need to find the force we need to apply to let the plane move with constant velocity, v.

What I am doing is this -

Take the plane as a system ( YES just the plane not including the sand getting attached to it ) It's rate of change in momentum is dP = (M+dm) v - Mv = vdm/dt which should be equal to the external force acting on the plane. Now the external force on the plane comes from the sand getting attached to it and the force which we need to apply ( F_us) So, $ F_{tot} = F_{us} + F_{dust} = vdm/dt$

Now F_(dust) is the force applied by dust on plane, so it is negative of force applied by plane on dust. So force applied by plane on dust is rate of change of dust's momentum which is (v-u) dm/dt. So force applied by dust is (u-v) dm/dt

So $F_{us} = (2v-u) dm/dt$...which is wrong because...

If dust applies force (v-u) dm/dt on plane then to allow the plane to move with a constant velocity, by newton's 2nd law we need to apply the force F_{us} = (v-u) dm/dt..

So what have I done wrong. I know if I take the system as dust + plane then I will get the correct answer but why am I not getting the answer when I take the system as ONLY THE PLANE after all I have accounted for all the external forces on the plane too.Then what is wrong.

I have found @Jan Lalinsky's answer Second law of Newton for variable mass systems regarding some sort of the same issue. Could anyone enlighten what is wrong here.

Shashaank
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    You can define your system as just the plane without the dust. But this system maintains constant mass. In such case $F_{tot}=ma$ applies. Since velocity of the system is constant in time, net force acting on this system has to be zero. In your notation, $F_{tot} = F_{us} + F_{dust} = 0$. – Ján Lalinský May 17 '20 at 11:02
  • @JánLalinský ok yeah thanks understood that... In $F_{tot} = ma$ what will be m... Just the mass of the plane right... So what I was doing wrong was INCLUDING that bit of mass of dust when it gets stuck to the plane in the plane system as well. The moment I did that my system changed from plane to plane + a bit of dust and because in the notation F= dp/dt m should be constant, I went wrong. Also I then again overdid and included that mass of dust again in just the dust system, essentially violating conservation of mass, so I went wrong. Could you pls let me know if this understanding iscorrect – Shashaank May 17 '20 at 11:17
  • Seems you've got it now. – Ján Lalinský May 17 '20 at 11:52
  • @JánLalinský thanks for the help – Shashaank May 17 '20 at 11:57

2 Answers2

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In general, the way in which we analyze variable mass systems is not by directly analyzing them but treating the system whose mass is changing and the small amount of mass getting added (or going out) as a system. So, essentially we are left with analyzing a system whose mass remains constant.

Now, coming to the method that you suggested. The basic flaw in your argument is violation of conservation of mass. While applying impulse-momentum theorem over the bigger chunk of mass (the plane) you assumed the initial mass to be M and the final mass as M+dm. You simultaneously applied the impulse-momentum theorem over the smaller mass (sand particle) and considered it's mass to be constant in that interval of time. So, it's initial mass was dm and final mass was dm as well.

The way you have applied the equations suggests that finally the mass of the system(Big chunk + small chunk) is M+2dm. From where did this extra mass dm come from?

I hope I have answered your question.

Suven Jagtiani
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  • Could you point out the exact step in my logic which you say is failing mass conservation probably as an edit – Shashaank May 16 '20 at 19:02
  • I am referring to the steps in which you wrote the small change in momentum of the plane(The first equation in your description) and the step in which you calculated F_sand by treating that sand particle as the system. – Suven Jagtiani May 17 '20 at 02:13
  • I have accepted the answer... I completely missed this simple point – Shashaank Jun 04 '20 at 11:57
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We need to apply some greater force than the sand because mass of the plane is increased after the sand strikes and so to make both of them move with same speed we need to apply some greater force. Also the change in momentum of sand will be

= ( V + u) dm because u was in opposite direction.