Find the force needed to accelerate body to a certain velocity for a certain time with respect to drag force

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.

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.

• 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. – user81619 Jun 21 '15 at 22:15
• 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." – Gonenc Jun 21 '15 at 22:30
• Hm, I don't know why it's rendering not properly... gonenc, sorry, I wrote this incorrect, was meant: "quadratic air resistance". Corrected now – gremlin Jun 21 '15 at 22:35
• 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? – garyp May 30 '16 at 18:56

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:

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

• 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. – gremlin Jun 21 '15 at 23:11
• 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. – Yukterez Jun 21 '15 at 23:21
• 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. – gremlin Jun 21 '15 at 23:22
• Drag will be ma=−cv^2, v is unknown. How can we solve this? – gremlin Jun 21 '15 at 23:24
• 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 – Yukterez Jun 21 '15 at 23:40