What's 'force per second'?

A force is an "instant" phenomenon so to speak. It might depend on time and be applied for a certain period of time. It may do work, such as your airplane thrust, and you can find the energy it transfers as $$W(t_0,t_1)=int_{t_0}^{t_1} vec F(t) cdot vec v(t) ;dt$$ At each instant, an infinitesimal amount of work $vec F(t)cdot vec v(t),Delta t$ is being added to the airplane's kinetic energy, so $vec F cdot vec v$ has units of J/s.

Similarly, you can construct something with units of N/s by taking the time derivative of a force. That would be its rate of change; i.e., in a time $Delta t$, a force would increase by $$vec F(t+Delta t) approx vec F(t)+Delta vec F = F(t)+frac{dvec F}{dt}Delta t$$ So you can think of the rate of change in N/s as a small amount of force being added each second; but what acts on objects is the force, not its rate of change.

It does not make sense to discuss a force with units of $rm N/s$ since those are not the units of force. The physical quantity that has these units is called yank and refers to the derivative of force with respect to time.

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An airplane's engine thrust is simply given as a force, but this must be a force applied by the engines each second (N/s)?

To answer this, I ask why you think the force should be applied every second, rather than some other time interval, such as every fortnight or Planck time? The problem here is that, in order for a finite force to have an effect, it must act over some non-zero duration. We see this in the equation for impulse. $$vec J = int_{t_1}^{t_2}vec F mathrm{d}t$$ Since force is inherently related to a rate of change, by Newton's Second Law, you are naturally inclined to think that its units should have seconds in the denominator. However, the actually already do, since the newton is defined by $$rm N = kgcdot m/s^2 = frac{kgcdot m/s}{s}$$ Note here that $rm kgcdot m/s$ is the unit of momentum and that net force is the time derivative of momentum.

As per second Newton law force is defined as momentum change over time : $$ F = dot p $$

If force is constant, then each second it will remain the same, thus defining "force per second" doesn't make much sense. However force can be function of time too, i.e. $F=F(t)$, then one can calculate force change over time : $$ frac {dF}{dt} = ddot p $$

Then total force acted over time interval $t_2-t_1$ is :

$$ F_{~tot} = int_{t_1}^{t_2} ddot p ~dt $$

If force increases monotonically and linearly, then one can define force per time period, or unit force :

$$ {F}_{1s},~[N/s] = frac {F_{tot}}{t_2-t_1} $$

Multiply that over time period elapsed and you will get force acting at that current time moment. That would be a closest interpretation of "force per second".