Engineering Mechanics: airplane landing

In summary,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.
  • #1
Joe_1234
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0
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.
 
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  • #2
$\dfrac{dv}{dt} = -k\sqrt{v}$ , 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.
 
  • #3
skeeter said:
$\dfrac{dv}{dt} = -k\sqrt{v}$ , 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.
 
  • #4
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.
 
  • #5
Joe_1234 said:
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.
 
  • #6
skeeter said:
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😊
 
  • #7
skeeter said:
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.
Sir, please help me how to get v(t)
 
  • #8
$\dfrac{dv}{dt} = -0.2 \sqrt{v}$

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

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

can you finish from here?
 
  • #9
Joe_1234 said:
Sir, please help me how to get v(t)

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...
 
  • #10
skeeter said:
$\dfrac{dv}{dt} = -0.2 \sqrt{v}$

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

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

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!
 
  • #11
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 ...

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

FAQ: Engineering Mechanics: airplane landing

1. How does an airplane land safely?

An airplane lands safely by using the principles of engineering mechanics. The pilot controls the speed and direction of the plane using the control surfaces, such as the flaps and ailerons, to adjust the lift and drag forces acting on the plane. The landing gear also plays a crucial role in absorbing the impact of the landing and providing stability.

2. What factors affect the landing of an airplane?

There are several factors that can affect the landing of an airplane, including wind speed and direction, weight and balance of the aircraft, runway conditions, and pilot skill. These factors must be carefully considered and accounted for in order to ensure a safe and successful landing.

3. How do engineers design airplane landing gear?

Engineers use a combination of mathematical calculations, computer simulations, and physical testing to design airplane landing gear. They consider factors such as the weight and size of the aircraft, the type of terrain the plane will be landing on, and the expected impact forces during landing. The goal is to design a landing gear that can safely support the weight of the plane and absorb the impact of landing.

4. Why do airplanes use flaps during landing?

Flaps are used during landing to increase the lift and drag forces on the wings of the airplane. This allows the plane to fly at a slower speed and descend at a steeper angle, making for a smoother and safer landing. Flaps also help to reduce the amount of runway needed for landing.

5. How does the shape of an airplane's wings affect its landing?

The shape of an airplane's wings, also known as the airfoil, plays a critical role in its landing. The shape of the airfoil determines the amount of lift and drag forces that can be produced, which affects the plane's speed and angle of descent during landing. Engineers carefully design the airfoil to optimize these forces for safe and efficient landings.

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