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

• songoku
In summary: It's now clear to me.In summary, the conversation discusses the equation for conservation of momentum, where the initial momentum is assumed to be zero. The question asks about the correct form of the equation and whether the speed of ejected oxygen needs to be constant. The experts suggest using the average speed of the collection of oxygen molecules and clarifies that the momentum conservation equation does not require the speed to be constant.
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) . vmax - 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
I am not sure, sorry.

songoku said:
So the equation should be like this?

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

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 and Lnewqban
haruspex said:
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

songoku said:
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.

Lnewqban and songoku
Thank you very much Lnewqban and haruspex

Lnewqban

What is the average ejection speed of gas?

The average ejection speed of gas can vary depending on the type of gas, temperature, and pressure. However, on average, the ejection speed of gas can range from 1000 to 5000 meters per second.

How is the average ejection speed of gas measured?

The average ejection speed of gas is typically measured using a device called a gas flow meter. This device measures the volume of gas flowing through a specific area in a given amount of time, allowing for the calculation of the average ejection speed.

What factors can affect the average ejection speed of gas?

The average ejection speed of gas can be affected by various factors such as the type of gas, temperature, pressure, and the size and shape of the ejection nozzle. Other factors such as the presence of obstacles or turbulence can also impact the ejection speed.

Why is the average ejection speed of gas important in scientific research?

The average ejection speed of gas is important in scientific research as it can provide valuable information about the behavior and properties of gases. It is particularly useful in studying gas dynamics, propulsion systems, and fluid mechanics.

Can the average ejection speed of gas be increased?

Yes, the average ejection speed of gas can be increased by increasing the pressure, temperature, or nozzle size. However, there are limitations to how much the ejection speed can be increased, and it is important to consider safety and efficiency when making adjustments.

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