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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:

- 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.
- A Milikan oil drop experiment, but using some particle which undergoes alpha decay with a very short half-life.

Thoughts? Opinions? Sarcastic remarks?