Dynamics - Airplane stopping on a runway

In summary, the problem involves calculating the distance required for an airplane to reduce its speed from 300m/hr to 150km/hr using a braking parachute and turning off the engines. The drag force on the aircraft is represented by the equation D=k*(v^2) where 'k' is a constant. Newton's second law, f=ma, can be used to solve for the acceleration, but the challenge is in solving for the displacement as all variables are expressed in terms of velocity.
  • #1
auburntigers_dy
4
0

Homework Statement


an airplane has a mass of 5Mg has a touchdown speed of 300m/hr. At which instant the braking parachute is deployed and the power shut off. If the total drag on the aircraft varies with velocity as shown in the accompanying graph, calculate the distance x along the runway required to reduce the speed to 150km/hr. Aproximate the variation of the drag by an equation D=k*(v^2). where 'k' is a constant.



Homework Equations


integral of (vdv)=a* integrate ds



The Attempt at a Solution


The constant 'k' was found to be 120,000 Newtons/(83m/s)^2
from the graph. Then, how to calculate the displacement is where the help is needed. The calculated answer from the back of the of the book is 201m. Any help on this problem would be appreciated.
thanks


 

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  • #2
I see no "attempt at a solution". Are you not even trying?
 
  • #3
auburntigers_dy,

I don't think that the "relevant equation" you cited is all that relevant. Can you write down Newton's second law for this problem? That would be a good start.
 
  • #4
tom mattson,
Newton's second law is f=m*a. ma=kv^2. therefore a=(kv^2)/m. This is where the relevant equation that i cited could be used, where the acceleration is calculated and velocity is given which then leaves you with the unknown term delta x. I am stuck on what velocity would i use to solve for this delta x.
 
  • #5
dynamics

HallsofIvy said:
I see no "attempt at a solution". Are you not even trying?


Hallsofivy,

i solved for k, the drag force is kv^2 which i equated it to Newton's second law Mass*acceleration. Then solving for acceleration resulted in k*v^2/(5000kg). then I integrated the relevant the relevant equation vdv=ads to solve for ds. The confusion lies in the calculated acceleration which is also expressed in terms of velocity which leaves stuck upon how to solve for displacement.
 
  • #6
dynamics

HallsofIvy said:
I see no "attempt at a solution". Are you not even trying?


Hallsofivy,

i solved for k, the drag force is kv^2 which i equated it to Newton's second law Mass*acceleration. Then solving for acceleration resulted in k*v^2/(5000kg). then I integrated the relevant equation vdv=ads to solve for ds. The confusion lies in the calculated acceleration which is also expressed in terms of velocity which leaves me stuck upon how to solve for displacement. everything is expressed in terms of velocity.
 

1. How does the weight of an airplane affect its stopping distance on a runway?

The weight of an airplane is a critical factor in determining its stopping distance on a runway. The heavier the airplane, the more kinetic energy it has, resulting in a longer stopping distance. This is why it is important for airplanes to be loaded with the correct weight and balance.

2. What role does the friction coefficient of the runway play in an airplane's stopping distance?

The friction coefficient of the runway is a measure of how much grip the surface has. This is a crucial factor in determining an airplane's stopping distance, as a higher friction coefficient means the airplane can generate more braking force and come to a stop in a shorter distance.

3. How does the speed of an airplane affect its stopping distance?

The speed of an airplane is a major factor in its stopping distance on a runway. The higher the speed, the more kinetic energy the airplane has, which means it will take longer to come to a stop. This is why pilots must carefully manage their speed during landing to ensure a safe and timely stop on the runway.

4. Can weather conditions affect an airplane's stopping distance on a runway?

Yes, weather conditions can have a significant impact on an airplane's stopping distance on a runway. In wet or icy conditions, the runway's friction coefficient is reduced, making it more difficult for the airplane to generate enough braking force to stop in a shorter distance. This is why pilots must take into account weather conditions when determining their approach and landing speed.

5. How do the brakes of an airplane work to slow it down on a runway?

The brakes of an airplane work by applying pressure to the wheels, causing friction between the tires and the runway. This friction converts the airplane's kinetic energy into heat, slowing it down. The brakes are controlled by the pilot through the use of pedals, similar to those in a car. In addition to the brakes, airplanes also use other methods such as reverse thrust and spoilers to help slow down during landing.

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