Airplane decelerating as a function of speed

Question

So, I have a problem where an airplane is decelerating as a function of speed. The acceleration is described as $a=dv/dt=-0.0035v^2-3$ as a function of time. For $t=0, v=83.3$ m/s. Can someone help me solve this, to figure out the function for the speed?

Thank you in advance

Answer 1

We know that, velocity is the first derivative of displacement wrt time, accln is the second derivative.

You know the expression for a=(dv/dt) . Rearrange and integrate with appropriate limits to find velocity as function of time.

Answer 2

You want to integrate to find speed as function of time $t$

$$ t = \int\frac{dv}{a} + c. $$

Answer 3

$$\frac{dv}{0.0035v^2+3}=-dt,$$ then

$$12.086\arctan(0.0276887v)=C-t.$$

With the given initial condition,

$$C=12.086\arctan(0.0276887\cdot83.3)\approx14.040.$$

Then

$$v\approx36.116\tan(0.08274(14.040-t)).$$

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The pilot should better switch off thrust reversal in $14$ seconds.

Answer 4

Hint

You have $$\frac{dv}{dt}=-av^2-b$$ As Narasimham answered, the easiest way is to rewrite the equation as $$\frac{dt}{dv}=-\frac 1{av^2+b}$$ So $$\int dt=-\int\frac {dv}{av^2+b}$$ Change variable $v=\sqrt {\frac b a}z$, $dv=\sqrt {\frac b a}dz$. This makes $$\int dt=-\frac 1{\sqrt{ab}}\int\frac{dz}{1+z^2}$$ which is a known integral.