Finding max. velocity of jet w/ constant and/or changing acceleration

1. May 15, 2013

mgsintheta

1. The problem statement, all variables and given/known data

You must evaluate the relationship between fuel consumption and the attainable velocity of a new jet fighter. Given that:

Force of jet= A(r(t))4/3
A: constant determined by the fighter model in the class being considered and the drag force on the plane
r(t): the rate of fuel consumption as a function of time

Consider 3 possible situations for r(t):

1. when the rate is constant for the duration of the acceleration period
2. when the rate is steadily decreasing for the duration of the acceleration period
3. when the rate decreases at a decreasing rate as the plane accelerates

For each of these you should assume that the rate is initially 500 ffu (fighter fuel units)/min,
and for each it is reasonable to suppose an acceleration period of 10 minutes.

Determine-in terms of the characteristics of the fighter (m and A)- the maximum velocity obtained by the jet

2. Relevant equations

F=ma

3. The attempt at a solution

If fjet=A(r(t))4/3,
then the Accelerationjet=[A(r(t))4/3]/m
-make A/m=C, then Accelerationjet=C[r(t)]4/3

Knowing that the integral of an acceleration should be the velocity function:
∫C[r(t)]4/3=C∫[r(t)]4/3→ C[(3/7)r(t)7/3+A]

My problem with the above is that I am not sure what to do about the constant. Does it matter? No initial conditions are given, so I don't believe that I can solve for it. If I was able to, it might be able to help me solve this problem.
Here is my dilemma. I understand that for situation 1, the acceleration is constant, and that I can use the constant-acceleration kinematics equation vf=vo+at. Since the acceleration is constant, the maximum velocity will be attained at the end of the acceleration period: 10C[550]4/3=45062.66988(A/m) However, for situations two and three, I am unsure of how to express the rate at which r(t) is decreasing, in order to be able to come up with a function that will give me the maximum velocity.
I understand that for situations 2 and 3, the maximum velocity will still be obtained at the end of the acceleration period. However, I am unsure of how to represent those answers differently than situation 1, as r(t) is constantly changing. I know that the graph of the acceleration vs. time graph for situation 2 is a straight diagonal line going down from left to right, and the graph for situation 3 is a concave up graph decreasing from left to right. Do I need to make an assumption about the rate of decrease for each of these situations? As in come up with a function to represent r(t) through the acceleration period? Or is there a way to come up with the maximum velocity concretely? I don't want the answer obviously, just some guidance to steer me towards that "aha" moment. Thanks for the assistance.
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. May 15, 2013

mukundpa

If the fuel is consumed whether the mass of the jet remains constant ?

3. May 16, 2013

davidchen9568

In the case when mass changes (because of fuel), F=ma no longer helps. Try F=dP/dt

4. May 16, 2013

mgsintheta

what exactly is P?

5. May 16, 2013

haruspex

In F = dp/dt, p stands for momentum. But I disagree with davidchen9568. You can still use F=ma, provided you bear in mind that m is variable. Indeed, F = dp/dt can lead you astray when the mass is varying.