Thrust

The power needed to generate thrust and the force of the thrust can be related in a non-linear way. In general, ${\displaystyle \mathbf {P} ^{2}\propto \mathbf {T} ^{3}}$. The proportionality constant varies, and can be solved for a uniform flow, where ${\displaystyle v_{\infty }}$ is the incoming air velocity, ${\displaystyle v_{d}}$ is the velocity at the actuator disc, and ${\displaystyle v_{f}}$ is the final exit velocity:

${\displaystyle {\frac {\mathrm {d} m}{\mathrm {d} t}}=\rho A{v}}$

${\displaystyle \mathbf {T} ={\frac {\mathrm {d} m}{\mathrm {d} t}}{v_{f}-v_{\infty }},{\frac {\mathrm {d} m}{\mathrm {d} t}}=\rho Av_{d}}$

${\displaystyle \mathbf {P} ={\frac {1}{2}}{\frac {\mathrm {d} m}{\mathrm {d} t}}(v_{f}^{2}-v_{\infty }^{2}),\mathbf {P} =\mathbf {T} v_{d}}$

Solving for the velocity at the disc, ${\displaystyle v_{d}}$, we then have:

${\displaystyle v_{d}={\frac {1}{2}}(v_{f}-v_{\infty })}$

When incoming air is accelerated from a standstill – for example when hovering – then ${\displaystyle v_{\infty }=0}$, and we can find:

${\displaystyle \mathbf {T} ={\frac {1}{2}}\rho A{v_{f}}^{2},\mathbf {P} ={\frac {1}{4}}\rho A{v_{f}}^{3}}$

From here we can see the ${\displaystyle \mathbf {P} ^{2}\propto \mathbf {T} ^{3}}$ relationship, finding:

${\displaystyle \mathbf {P} ^{2}={\frac {\mathbf {T} ^{3}}{2\rho A}}}$

The inverse of the proportionality constant, the "efficiency" of an otherwise-perfect thruster, is proportional to the area of the cross section of the propelled volume of fluid (${\displaystyle A}$) and the density of the fluid (${\displaystyle \rho }$). This helps to explain why moving through water is easier and why aircraft have much larger propellers than watercraft.

A very common question is how to compare the thrust rating of a jet engine with the power rating of a piston engine. Such comparison is difficult, as these quantities are not equivalent. A piston engine does not move the aircraft by itself (the propeller does that), so piston engines are usually rated by how much power they deliver to the propeller. Except for changes in temperature and air pressure, this quantity depends basically on the throttle setting.

A jet engine has no propeller, so the propulsive power of a jet engine is determined from its thrust as follows. Power is the force (F) it takes to move something over some distance (d) divided by the time (t) it takes to move that distance:[7]

${\displaystyle \mathbf {P} =\mathbf {F} {\frac {d}{t}}}$

In case of a rocket or a jet aircraft, the force is exactly the thrust (T) produced by the engine. If the rocket or aircraft is moving at about a constant speed, then distance divided by time is just speed, so power is thrust times speed:[8]

${\displaystyle \mathbf {P} =\mathbf {T} {v}}$

This formula looks very surprising, but it is correct: the propulsive power (or power available [9]) of a jet engine increases with its speed. If the speed is zero, then the propulsive power is zero. If a jet aircraft is at full throttle but attached to a static test stand, then the jet engine produces no propulsive power, however thrust is still produced. The combination piston engine–propeller also has a propulsive power with exactly the same formula, and it will also be zero at zero speed – but that is for the engine–propeller set. The engine alone will continue to produce its rated power at a constant rate, whether the aircraft is moving or not.

Now, imagine the strong chain is broken, and the jet and the piston aircraft start to move. At low speeds:

The piston engine will have constant 100% power, and the propeller's thrust will vary with speed
The jet engine will have constant 100% thrust, and the engine's power will vary with speed

If a powered aircraft is generating thrust T and experiencing drag D, the difference between the two, T − D, is termed the excess thrust. The instantaneous performance of the aircraft is mostly dependent on the excess thrust.

Excess thrust is a vector and is determined as the vector difference between the thrust vector and the drag vector.

The thrust axis for an airplane is the line of action of the total thrust at any instant. It depends on the location, number, and characteristics of the jet engines or propellers. It usually differs from the drag axis. If so, the distance between the thrust axis and the drag axis will cause a moment that must be resisted by a change in the aerodynamic force on the horizontal stabiliser.[10] Notably, the Boeing 737 MAX, with larger, lower-slung engines than previous 737 models, had a greater distance between the thrust axis and the drag axis, causing the nose to rise up in some flight regimes, necessitating a pitch-control system, MCAS. Early versions of MCAS malfunctioned in flight with catastrophic consequences, leading to the deaths of over 300 people in 2018 and 2019.[11][12]