Non-intuitive velocity change for object with changing mass

[Dishsoap]

Greetings, fellow PFers,

I have an interesting question which one of my professors raised, and I'm intrigued and would like to try it experimentally, though how to do so isn't clear to me.

Suppose a cart with a constant force on it (say 1N), and with a velocity v(0)=-10 and a mass that changes according to $m(t)=e^{-t}$. That is, the cart is initially moving in a direction opposite the force. Therefore, the equation of motion gives

$F=\frac{d}{dt} m v = m \frac{dv}{dt} + v \frac{dm}{dt}$
or, solved with the method of integrating factors,
$v(t)=e^{t}(t-10)$

Which, to visualize, gives something like this:

That is, the speed (not velocity) actually increases, before decreasing and the car eventually turning around.

This is intriguing and I would really like to see this experimentally, however the difficulty is avoiding "shooting off" the mass in such a way as to create recoil. Also, an exponentially decaying mass will be hard to realize, but I will frankly settle for any mass function which decreases monotonically. I have had a couple of ideas:

1. A cart that gradually leaks water. However, the only way I can think to do this is to use the classic photogate/pulley system used for physics 1, which I think would not be accurate enough for this purpose.
2. A Milikan oil drop experiment, but using some particle which undergoes alpha decay with a very short half-life.

Thoughts? Opinions? Sarcastic remarks?

 

[Student100]

Interesting, instead of pulleys you could develop a circular air track, enclosed in a weak vacuum chamber with a modified rider holding water. You could then take advantage of the water to gaseous transformation to model the mass loss. Getting the escape to be a variable rate might be a challenge in design, and taking your measurements could be challenging.

 

[andrewkirk]

Regarding a practical implementation: wouldn't the mass of a sublimating hemispherical block of dry ice decay exponentially?

Regarding the equations, I don't think the equation is necessarily correct. It assumes that all momentum is conserved in the remaining mass, which implies that the lost mass has zero momentum at the instant it detaches. It is that phenomenon that gives you your initially increasing speed, because the momentum is getting concentrated into a smaller and smaller mass.

In practice, it seems likely that, leaving aside the accelerating force, the body's momentum reduces in proportion to the loss of mass, because the detaching particles initially have the same velocity as the object, in the object's direction of motion. In that case the equation of motion is simpler: $\frac{dv}{dt}=\frac{1}{m(t)}=e^t$. Hence $v=v(0)+e^t$ and displacement $x=-10t+e^t$. In this case the speed decreases right from the beginning until it reaches zero, then heads off in the opposite direction exponentially.

 

[A.T.]

That is, the speed (not velocity) actually increases, before decreasing and the car eventually turning around.

Choose the reference frame where the cart is initially at rest. You have a force in positive direction, and want the cart to accelerate in the opposite direction? The only way to do this, is to shoot off mass forward from the cart, which will create thrust just like a rocket engine.

As andrewkirk shows, It will not work by just dropping the mass with zero relative horizontal velocity.

 

[A.T.]

It assumes that all momentum is conserved in the remaining mass, which implies that the lost mass has zero momentum at the instant it detaches.

It basically assumes a rocket engine, with variable expulsion speed, controlled such that the expelled gas is stationary in the frame where v(0)=-10. Due to the decaying mass flow, the negative rocket thrust is initially higher, then lower than the constant positive force.

 

[Dishsoap]

Thank you all for your insight. So there is no real experiment in which I would be able to see this effect?

 

[Noctisdark]

$m = e^{-t}$ means that after while the mass is basically zero, this is impossible a better suggestion will be $m = m_0 + a*e^{-t}$ where $m_0$ is the mass of the container and $a$ is the initial mass of the water, however as I've seen above momentum will not be conserved, but even though if a is very small you can assume that is conserved but the effect you've mentionned cannot be true, $v = be^{t - ln(a +m_0e^t)} + \frac{Fe^t t}{a + me^t}$ which after a while will yield y = b + F*t/m, if you burn the container (which is a larger mass) momentum conservation will stop this from happening, :D Good luck and that was really interesting !

 

[andrewkirk]

So there is no real experiment in which I would be able to see this effect?

I presume that by This Effect you mean to refer to the pattern of movement shown in the graph. If so then yes there doesn't appear to be any such experiment without introducing very unnatural system controls such as computerised feedback. But that's because (I think) the graph is based on a mistake. I think that whoever thought up this situation wrote the equation that you solved without realising that it implied that all of the evaporating particles' momentum was magically transferred back to the unevaporated part of the object. I think they were probably just wondering what would be the equation of motion of an evaporating object, wrote that equation without realising that it implied that totally implausible assumption, then observed that it gave an interesting pattern of motion and wrongly concluded that the interesting pattern would be the real pattern of the simple experiment they imagined.

It's only because of the mistake that the pattern of motion is difficult to replicate. The initial idea was simple, and if pursued without the implausible assumption of momentum recapture, leads to an equation, shown in post #3, that should be easy enough to experimentally reproduce, but the mistake turned it into an equation of motion that is unrelated to the initial practical idea, and hence as hard to experimentally reproduce as any randomly chosen complicated function of time.

Having said that, even the original idea is not that easy to reproduce experimentally. I realized that my idea of a hemisphere of dry ice won't give the desired rate of mass loss because the proportional rate of shrinking will increase as it gets smaller, rather than remaining constant as required by $m=e^{-t}$ I think something a little more sophisticated would be needed. eg the dry ice could be in a container with an opening with a variable aperture. The size of the aperture could be controlled by a sliding shutter, whose position is related to the weight of the dry ice by some sort of spring and balance arrangement. The size of the aperture would determine the rate of evaporation. It shouldn't be too hard for a clever engineer to rig up, without needing to involve computer control.