So, the problem is straightworward when we suggest that air resistance force is constant:
$$ \vec F = \frac {\vec V_1 – \vec V_0} {t} m / b $$
$$ \vec V_0, \vec V_1 – \text {initial and final velocities respectively},\\ t – \text {time, during which velocity will become its final value}, \\b – \text {some constant for drag force} $$
The problem arises when using quadratic air resistance: $$m a = -c v^2$$
I don't have any clue for solving this and asking for help.
What I must do is to accelerate plane to the certain velocity in a certain period of time. So, I must find needed acceleration or force, itsn't matter, because can be easily recalculated.
Thanks in advance!
P.S. It's sad, but I'm not familiar with calculus, I'm more programmer, not mathematician.
P.P.S. I know what differentiating and integration is, what derivative and displacement is, but really not familiar with differential equations, but guess it is here.
Best Answer
If your object has mass $M$ and you want to accelerate it with acceleration $a$ to a specific end-velocity $v$ you have to keep in mind that the energy
$$e = \frac{M\cdot v^2}{2}$$
and also
$$e = F\cdot x$$
where $x$ is the distance over which the force $F$ which equals $M\cdot a$ is applied. Knowing that you can solve for the distance over which you have to accelerate:
$$\frac{M\cdot v^2}{2} = M\cdot a\cdot x \to $$
$$ \to x = \frac{v^2}{2\cdot a}$$
Now you have the distance and can easily solve for the time you need to reach your velocity $v$:
$$t = \sqrt{\frac{2\cdot x}{a}}$$
Example: If you have a drag force of $Fd = -4 \frac{kg \cdot m}{s^2}$, your object has a mass of $M = 3 kg$ and you want to reach $v = 10 m/s$ in $t = 5 s:
$$a = \frac{v}{t} \to a = \frac{10 m/s}{5 s} = 2 \frac{m}{s^2}$$
You multiply that with yor mass and get a force of
$$F = 6 \frac{kg \cdot m}{s^2}$$
This is the force you would need without drag force. Now you simply substract the drag:
$$F_{total} = F-Fd = (6+4)N = 10 N$$