# What is wrong with my approach to this BPHO problem

I am a tutor, and one of my students is practicing for the BPHO. One of the problems he wanted to go over is below.

I decided to use the fact that

$$F_{net}=\frac{dp}{dt}=M\frac{dv}{dt}+\frac{dM}{dt}v$$

for $$M$$ the total mass. From there, you can set up an ODE for $$v$$ and solve. However, in the official solutions for that problem, they instead set up their ODE as $$\frac{dv}{dt}=\frac{F_{net}}{M}$$

These ODEs have different (though similar) solutions, but I can't think of why my approach would be considered incorrect.

EDIT: To be clear, my issue here is why taking $$F_{net}=Ma$$, as the examiners did, is considered to be correct, while $$F_{net}=\frac{dp}{dt}$$ is not. Am I missing or forgetting some reason the latter would just simplify to the former?

• @WillO Yes, I am aware. Hence why I have a dM/dt term in that momentum derivative. Similarly, the official solution also has M as a function of time. Nov 5, 2022 at 17:31
• I misread what you were saying, so my comment is irrelevant. I am deleting it, with apologies. Nov 6, 2022 at 2:05
• I have voted to reopen. It seems to me there is an interesting conceptual question here. Nov 6, 2022 at 9:17
• I think your method is right, don't really see why there wouldn't be a $\dot m v$ term. I have heard from others that BPhO can sometimes be incorrect and this may be one of those problems (do take that with a grain of salt as I have never done any BPhO problem before). Nov 7, 2022 at 23:24
• Unless in the BPHO solution they didn't somehow put the $\dot{M}v$ term into their $F_{net}$ I would agree with you and Ashmit Dutta. Nov 8, 2022 at 11:59