Find power needed to fly this airplane using momentum considerations

[mmfiizik]

Homework Statement

Plane which flies at velocity v, every second takes m mass of air and consumes M mass of fuel. Combustion products are released at velocity u relative to the plane. Find power of the plane P.

Relevant Equations

Change in momentum = force x time

I just don't understand should I take u relative to the plane or relative to the ground.
I tried to solve it like this:
$$p_{final}=m_{0}v-m(u-v)-M(u-v)$$
$$p_{initial}=m_{0}v$$
$$\Delta p=-m(u-v)-M(u-v)$$
$m_0$ is mass of the plane.
$$F=\Delta p$$
$$F=-m(u-v)-M(u-v)=(m+M)(v-u)$$
$$P=Fv=(m+M)(v-u)v$$
Or should I write in the first equation velocity of combustion products just u?

 

[haruspex]

should I take u relative to the plane or relative to the ground.

Always worth checking a special case. What if the fuel were simply dumped instead of being burnt? What would u be? Do your equations give the right result?

There is an important difference between the fuel and the air. You have simply added them.

 

[erobz]

Since we are talking about rates here you should probably start with:

$$ F~dt = ( p+dp) - p $$

Where $p$ is the momentum of the system consisting of the planes mass ($M_p$), mass of fuel carried ($M$), and ejected mass air /fuel ($dm,dM_e$). The velocities of the various components are w.r.t. an inertial frame. $u$ is defined as relative to the plane so you must make that adjustment for components of the ejecta.

 

[jbriggs444]

The rest frame of the plane is a good one to use since it allows us to ignore the momentum change from the decreasing mass of the plane over time. Instead, we can concentrate on the momentum flux from the incoming air and from the outgoing exhaust.