# 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.

• MarkFL

#### 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

#### MarkFL

$$\displaystyle \d{v}{t} = -k\sqrt{v}$$ where $$v(0)=v_0$$
$$\displaystyle \int_{v_0}^{v(t)} \frac{1}{\sqrt{a}}\,da=-k\int_0^t\,db$$