Is This Momentum Conservation Equation Correct for a Gas Ejection Problem?

[songoku]

Homework Statement

A trolley has cylinder on top of it and inside the cylinder there is oxygen gas. The trolley and cylinder has mass of 0.68 kg and the initial mass of oxygen is 12 g. Calculate the average ejection speed of the oxygen gas if the maximum speed of trolley + cylinder is 2.7 m/s

Relevant Equations

Conservation of momentum

I assume the system starts from rest so the total initial momentum is zero.

Let:
M = mass of trolley + cylinder
m = initial mass of oxygen
Δm = mass of ejected oxygen
u = speed of ejected oxygen

Conservation of momentum:
0 = (M + m - Δm) . v_max - u . Δm

1) Is my equation correct?

2) I suppose I need to assume u is constant?

3) Do I need to know u to answer this question?

4) Is it correct that the question asking to find $\frac{\Delta m}{\Delta t}$

Thanks

 

[Lnewqban]

Please, see point #6:
https://www.physicsforums.com/insights/frequently-made-errors-mechanics-momentum-impacts/

The maximum speed of the trolley-cylinder should be reached after the cylinder is empty.

 

[songoku]

Please, see point #6:
https://www.physicsforums.com/insights/frequently-made-errors-mechanics-momentum-impacts/

The maximum speed of the trolley-cylinder should be reached after the cylinder is empty.

So the equation should be like this?

$$\vec F_{net}=\frac{M.v_{max}-m.u}{\Delta t}$$
$$0=\frac{M.v_{max}-m.u}{\Delta t}$$

And the question is asking about $u$? Thanks

 

[Lnewqban]

I am not sure, sorry.
Let's ask @haruspex

 

[haruspex]

So the equation should be like this?

$$\vec F_{net}=\frac{M.v_{max}-m.u}{\Delta t}$$
$$0=\frac{M.v_{max}-m.u}{\Delta t}$$

And the question is asking about $u$? Thanks

Yes, but you do not need to assume u constant.
The average speed is $\frac{\int u.dm}{\int dm}$, and the momentum conservation is $Mv=\int u.dm$.

 

[songoku]

Yes, but you do not need to assume u constant.
The average speed is $\frac{\int u.dm}{\int dm}$, and the momentum conservation is $Mv=\int u.dm$.

I am sorry I don't understand this. Why the average speed is $\frac{\int u.dm}{\int dm}$?

And for momentum conservation, $u$ is function of $m$ where $m$ is mass of oxygen ejected? So if I give the limit to the integration, it would be like this?
$$Mv=\int_{m_o}^{0} u.dm$$

Thanks

 

[haruspex]

I am sorry I don't understand this. Why the average speed is $\frac{\int u.dm}{\int dm}$?

And for momentum conservation, $u$ is function of $m$ where $m$ is mass of oxygen ejected? So if I give the limit to the integration, it would be like this?
$$Mv=\int_{m_o}^{0} u.dm$$

Thanks

As a mater of definition, the formula for the average of variable Y with respect to variable X is $\frac{\int Y.dX}{\int dX}$. The average velocity of a collection of masses is the average with respect to mass.

It is not a question of limits; it is the average over the collection of oxygen molecules.

 

[songoku]

Thank you very much Lnewqban and haruspex