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.

  • 2
    $\begingroup$ Hi just letting you know your post is not rendering properly on my tablet (might be a temporary issue) and you have a better chance of an answer if you show the details of any attempts you have tried yourself. Best of luck with it. $\endgroup$
    – user81619
    Jun 21, 2015 at 22:15
  • $\begingroup$ I really don't understand this sentence: "The problem arises when using quadratically drag force. I don't have any clue for solving this and asking for help." $\endgroup$
    – Gonenc
    Jun 21, 2015 at 22:30
  • $\begingroup$ Hm, I don't know why it's rendering not properly... gonenc, sorry, I wrote this incorrect, was meant: "quadratic air resistance". Corrected now $\endgroup$
    – gremlin
    Jun 21, 2015 at 22:35
  • $\begingroup$ Your description of $t$ is ... odd. So is the phrase "for a certain time". I don't know what you mean. It seems the problem is over specified, but I may not understand what you are saying. Is $t$ the duration that the force is acting? What do you know about the drag force? $\endgroup$
    – garyp
    May 30, 2016 at 18:56

1 Answer 1


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:

$$F_{total} = F-Fd = (6+4)N = 10 N$$

  • $\begingroup$ Thanks, but you didn't understand me correctly. Because t is known. Maybe I should give concrete example. What I must do is to accelerate plane to the certain velocity in the some period of time. So, I must find needed acceleration and start simulation. Maybe it became clear. $\endgroup$
    – gremlin
    Jun 21, 2015 at 23:11
  • $\begingroup$ In the example you find the way to calculate the force you have to put in when the time in which you have to reach your velocity is known. $\endgroup$
    – Yukterez
    Jun 21, 2015 at 23:21
  • $\begingroup$ Ha, yes, it is, but there is a problem, because Fd will vary on speed. This is actual problem. The main problem that further body's speed is unknown at start. $\endgroup$
    – gremlin
    Jun 21, 2015 at 23:22
  • $\begingroup$ Drag will be ma=−cv^2, v is unknown. How can we solve this? $\endgroup$
    – gremlin
    Jun 21, 2015 at 23:24
  • $\begingroup$ There is no analytical solution for this problem, if you want to have real air resistance you have to use the ballisic differential equation which is a litte more complicated and can only be solved numerically, see yukterez.net/ballistik/#plot $\endgroup$
    – Yukterez
    Jun 21, 2015 at 23:40

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