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Old Mar 25th 2014, 06:56 AM   #1
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Question about the rocket - exhaust speed...

I have a question to which I already answered, but I have impression that I'm not so clear as I should be.

Can someone just point out if something is wrong, please?

Question is as follows:

Noting that the exhaust speed in part (b) is 2.00 times higher than that in part (a), explain why the required fuel mass is not simply smaller by a factor of 2.00.

and my answer is:

"The final rocket's speed is NOT simply proportional to these two factors: the exhaust speed and the mass of fuel required.

The final speed of a rocket is directly proportional to the a)exhaust speed of the fuel and to the b)natural logarithm of the ratio [(M_i - mass of the rocket + initial mass of the fuel)/(M_f - mass of the rocket)]. Furthemore, the exhaust speed shows the "thrust" of the fuel - the capacity of the fuel to generate a certain speed.

That's why in the second case, when the exhaust speed is increased by a factor of 2, the amount of fuel decreases exponentially."
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Old Mar 25th 2014, 09:26 AM   #2
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Couple of things you should clarify:

1. Please define the terms M_i and M_f.
2. This appears to be a problem involving a rocket acting in space, so that air resistance is ignored - hence the rocket's acceleration is solely a function of engine thrust and the rockets mass as a function of time. I think you shoudl state that assumption.
3. How did you derive the equation for final speed of the rocket?
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Old Mar 25th 2014, 10:19 AM   #3
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Any increase in efficiency of a rocket motor has a more than proportional benefit in the final speed.

This is because the rate at which the rocket accelerates relative to the rate at which it looses mass, as the fuel burns, changes.

The effect is a bit like the compound interest calculations.
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Old Mar 25th 2014, 10:35 AM   #4
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Originally Posted by ChipB View Post
Couple of things you should clarify:

1. Please define the terms M_i and M_f.
2. This appears to be a problem involving a rocket acting in space, so that air resistance is ignored - hence the rocket's acceleration is solely a function of engine thrust and the rockets mass as a function of time. I think you shoudl state that assumption.
3. How did you derive the equation for final speed of the rocket?
I actually clarified the terms M_i and M_f in the brackets, as being [(M_i - initial mass of the rocket + initial mass of the fuel)/(M_f - final mass of the rocket)].

Thanks about the hint in 2.

Concerning the point 3, the final equation was actually derived directly in the book, considering the systems of the rocket and the ejected fuel as an isolated system, which after equating the initial and final momentum of it and integrating gives as the

v_f-v_i = v_e*ln (M_i/M_f) , I got all the steps of this equation (even if not actively deriving it), but why are you asking me this? ... to make sure I use the right argument(equation) for my question?

Thanks
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Old Mar 25th 2014, 10:37 AM   #5
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Originally Posted by MBW View Post
Any increase in efficiency of a rocket motor has a more than proportional benefit in the final speed.

This is because the rate at which the rocket accelerates relative to the rate at which it looses mass, as the fuel burns, changes.

The effect is a bit like the compound interest calculations.
Ok, so I can say I got it right(the explanaiting) even if its longer

Thanks
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Old Mar 25th 2014, 04:11 PM   #6
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Originally Posted by dokrbb View Post
I actually clarified the terms M_i and M_f in the brackets, as being [(M_i - initial mass of the rocket + initial mass of the fuel)/(M_f - final mass of the rocket)].
This is something you could improve in your answer. What you wrote is a fraction consisting of (M_i minus initial mass of rocket + initial mass of fuel)/(M_f minus final mass of rocket). I suggest you define these terms and then show the fraction:

Given M_i = initial mass of rocket + fuel and M_f = final mass of rocket after all fuel is expended, then "the final speed is proportional to ....(b) narural logarithm of ratio (M_i/M_f)."

Originally Posted by dokrbb View Post
but why are you asking me this? ... to make sure I use the right argument(equation) for my question?
I asked because I didn't know where you got it from.
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Old Mar 26th 2014, 05:40 PM   #7
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Originally Posted by ChipB View Post
This is something you could improve in your answer. What you wrote is a fraction consisting of (M_i minus initial mass of rocket + initial mass of fuel)/(M_f minus final mass of rocket). I suggest you define these terms and then show the fraction:

Given M_i = initial mass of rocket + fuel and M_f = final mass of rocket after all fuel is expended, then "the final speed is proportional to ....(b) narural logarithm of ratio (M_i/M_f)."



I asked because I didn't know where you got it from.
Thanks ChipB, you are right, it was a bit confusing to put all together in the equation,

Thanks again,
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