# Engineering Mechanics: airplane landing

#### Joe_1234

##### New member
The landing speed of an airplane is 360 kph. When it touches down, it puts on its brakes and reverses its engines. The retardation in its speed is 0.2 times the square root of its speed. Determine the time elapsed in seconds from the point of touchdown until the plane comes to a complete stop.

#### skeeter

##### Well-known member
MHB Math Helper
$\dfrac{dv}{dt} = -k\sqrt{v}$ , where $k=0.2$

solve the separable differential equation for velocity as a function of time.

you are given $v_0 = 360 \, km/hr$ (you’ll need to convert to m/s). use it to determine the constant of integration.

#### Joe_1234

##### New member
$\dfrac{dv}{dt} = -k\sqrt{v}$ , where $k=0.2$

solve the separable differential equation for velocity as a function of time.

you are given $v_0 = 360 \, km/hr$ (you’ll need to convert to m/s). use it to determine the constant of integration.
Sir thank you.

#### Joe_1234

##### New member
Thank you Sir. Please help me solve again for the length of runway it will from the point of touchdown until it comes to a complete stop.

#### skeeter

##### Well-known member
MHB Math Helper
Thank you Sir. Please help me solve again for the length of runway it will from the point of touchdown until it comes to a complete stop.
once you've determined $v(t)$ and solved for the time required to stop ...

$\displaystyle \Delta x = \int_0^T v(t) \, dt$ , where $T$ is the time required to come to a full stop.

#### Joe_1234

##### New member
once you've determined $v(t)$ and solved for the time required to stop ...

$\displaystyle \Delta x = \int_0^T v(t) \, dt$ , where $T$ is the time required to come to a full stop.
Thanks a lot sir

#### Joe_1234

##### New member
once you've determined $v(t)$ and solved for the time required to stop ...

$\displaystyle \Delta x = \int_0^T v(t) \, dt$ , where $T$ is the time required to come to a full stop.

#### skeeter

##### Well-known member
MHB Math Helper
$\dfrac{dv}{dt} = -0.2 \sqrt{v}$

$\dfrac{dv}{\sqrt{v}} = -0.2 \, dt$

$2\sqrt{v} = -0.2t + C$

can you finish from here?

#### MarkFL

Staff member
As skeeter posted, finding the velocity involves solving the following IVP:

$$\displaystyle \d{v}{t} = -k\sqrt{v}$$ where $$v(0)=v_0$$

The ODE associated with this IVP is separable, and I would next write:

$$\displaystyle \int_{v_0}^{v(t)} \frac{1}{\sqrt{a}}\,da=-k\int_0^t\,db$$

Try seeing if you can proceed from there and get the same result you get from skeeter's post made above just now...

#### Kyosh

##### New member
$\dfrac{dv}{dt} = -0.2 \sqrt{v}$

$\dfrac{dv}{\sqrt{v}} = -0.2 \, dt$

$2\sqrt{v} = -0.2t + C$

can you finish from here?
Hello! If it's not too much to ask, can you please dumb this down for me (aka someone who's good with Physics concepts but absolute trash with Mathematics)? Thank you. I hope you have a nice day!

#### skeeter

##### Well-known member
MHB Math Helper
The problem is rather straightforward ... you are given an initial speed, an acceleration as a function of speed, and a final speed.

note $v_0 = 360 \text{ km/hr } = 100 \text{ m/s}$ and $v_f = 0$

$a = \dfrac{dv}{dt} = -0.2 \sqrt{v}$

separating variables yields ...

$v^{-1/2} \, dv = -0.2 \, dt$

integrating both sides ...

$2v^{1/2} = -0.2t + C$, where $C$ is a constant of integration

$v_0 = 100 \text{ m/s } \implies C = 20 \implies v = (10 - 0.1t)^2$

$v_f = 0 \implies t = 100 \text{ s}$

to get the distance the plane travels before it comes to a stop ...

$\displaystyle D = \int_0^{100} (10 - 0.1 t)^2 \, dt \approx 3333 \text{ m}$