Conservation of Momentum With Changing Mass

[Fascheue]

Homework Statement

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A rocket that starts at rest with mass M ejects exhaust at a given speed u. What is the mass of the rocket (including unused fuel) when its momentum is maximum? What is the mass when its energy is maximum?

Homework Equations

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p = mv

The Attempt at a Solution

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I’m not really sure to start with this one. I know that the change in momentum of the rocket is equal in magnitude and opposite in direction to the change in momentum of the fuel. I’d imagine I will also be looking for when dp/dt is equal to 0 to find the maximum momentum.

 

[berkeman]

I’d imagine I will also be looking for when dp/dt is equal to 0 to find the maximum momentum.

You will have to do the math to figure out when the momentum is maximum. Are you given the mass of the fuel too, or just the total M?

You may have a larger momentum before all the fuel is used up, since the momentum and energy depend on the mass of the rocket plus unused fuel...

 

[Ray Vickson]

Homework Statement

[/B]
A rocket that starts at rest with mass M ejects exhaust at a given speed u. What is the mass of the rocket (including unused fuel) when its momentum is maximum? What is the mass when its energy is maximum?

Homework Equations

[/B]
p = mv

The Attempt at a Solution

[/B]
I’m not really sure to start with this one. I know that the change in momentum of the rocket is equal in magnitude and opposite in direction to the change in momentum of the fuel. I’d imagine I will also be looking for when dp/dt is equal to 0 to find the maximum momentum.

Google "rocket equation" for related material.

 

[Fascheue]

Google "rocket equation" for related material.

Should I be using this equation? I’m not quite sure how I can do that without knowing what Δv is when momentum is at its maximum value.

 

[Ray Vickson]

Should I be using this equation? I’m not quite sure how I can do that without knowing what Δv is when momentum is at its maximum value.

That is not what I meant. If you Google "rocket equation" you should find entire articles or lectures on the topic, where the author writes out the differential equations of motion in detail, and then eventually solves them. The derivations are likely to be more useful than a single formula such as the one you wrote above, because your question requires that you know the velocity and the mass of the rocket at each point in time, in order to allow you to maximize the rocket's momentum.

 

[Fascheue]

I used the equation V = uln(M/M-m) where M is the initial mass and m is the mass of the projectile. I then got P = (M-m)uln(M/M-m) by multiplying each side by the total mass of the rocket. I then found the derivative with respect to m and am now trying to find where it is equal to zero.So far I have this, but I’m not quite sure how to solve it (if it’s even right).

2ln(M/(M-m) = (M+m)/(M^2-Mm)

 

[berkeman]

I used the equation V = uln(M/M-m) where M is the initial mass and m is the mass of the projectile. I then got P = (M-m)uln(M/M-m) by multiplying each side by the total mass of the rocket. I then found the derivative with respect to m and am now trying to find where it is equal to zero.So far I have this, but I’m not quite sure how to solve it (if it’s even right).

2ln(M/(M-m) = (M+m)/(M^2-Mm)

Could you please show your work in detail so we can check it? Also, it helps if you post in LaTeX so the math symbols are more traditional instead of us trying to decode a clear text version of your math work. There is a LaTeX tutorial under INFO at the top of the page when you click on Help/How-To. Thanks.

EDIT / ADD -- Also, if you could post a link to the rocket equation that you found via Google, that would help to keep us all on the same page. Thanks.