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Hoping somebody could give me a hand with this. Unfortunately I haven't taken physics yet, my background is in math/computer science.

I'm studying computational science and the problem is essentially asking what is the force needed to maintain the velocity of a train with while its raining? (water is accumulating, mass of train is increasing).

I'm having trouble figuring out how to relate the changing mass.

In computational science I'm working on this problem as a matrix representation: $$ F = ma \quad \rightarrow \quad f-Ku=Mu''$$ Where $u$ is the displacement vector. $M$ is the mass matrix. $f$ is the external force vector and $Ku$ the internal force vector.

Additional information: The rate the mass is changing is $r= \frac{dm}{dt}$

I'm not looking for just the answer; this is driving me nuts, I should be able to figure it out. But its just not been going well.

-------To break it down--------

train $T$ is traveling at a constant velocity $v$. Mass is a function of time and displacement. The force $F = ma = \frac{d}{dt}(m(u,t)\frac{du}{dt})= \frac{dm}{dt}\frac{du}{dt}$

I am looking for where $ma = 0 \text{ and } F \neq 0 $

Anyways, I'm hoping somebody could give me a nudge here, I don't know how I can represent this properly. Thanks

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    $\begingroup$ The equation $F_{total}=ma$ is a special case of the more general $F_{total}= \frac{d(mv)}{dt} = m\frac{dv}{dt}+v\frac{dm}{dt}$ $\endgroup$
    – danimal
    Commented May 8, 2015 at 0:21
  • $\begingroup$ Why is the increasing mass relevant here? The force is only necessary to accelerate the accumulating rain to the speed of the car. The mass of the car doesn't matter. $\endgroup$
    – BowlOfRed
    Commented May 8, 2015 at 0:30
  • $\begingroup$ @BowlOfRed Mass increases with velocity, a computational model would have diminished accuracy. $\endgroup$
    – spun
    Commented May 8, 2015 at 0:41
  • $\begingroup$ Assuming a steady velocity and steady rainfall, the rain quickly accumulates to a uniform sheen of water which covers the moving train. Mass of the moving train doesn't increase any further. So, you shouldn't have to deal with dm/dt. $\endgroup$
    – Ernie
    Commented May 8, 2015 at 0:51
  • $\begingroup$ My comment is not right. Acceleration isn't instantaneous, so you need t to get the train back up to V after the rain covers it. $\endgroup$
    – Ernie
    Commented May 8, 2015 at 1:01

1 Answer 1

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Use the fact that force is equal to the rate of change of momentum. The rain is accelerated to the same velocity as the train, so if you know the mass of rain per second you can calculate the momentum change per second. This is equal to the force required.

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