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# Physics/Mechanics

Question
A ship, on which airplanes land, has an airstrip of 70 m. The ship uses three different techniques to stop airplanes when they are landing.
The first technique is to use a cable reel of radius 0.4 m, around which a non-elastic rope is coiled. The reel is to be rotated about its axle - which is a fixed point (doesn't move) - opposite to a friction force cause by two brake pads in contact with the outer radius of the reel that is 0.6 m. The coefficient of static and kinetic friction between a brake pad and the reel are 0.45 and 0.41, respectively.

The second technique is to use an elastic rope that restrains the airplane in a way that it exerts a force, on the airplane, proportional to the rope extension.

The third technique is to use a piston - located inside a cylindrical water tank which is lying horizontally - such that when an airplane lands, the piston pumps water out, through a perforation at the end of the tank, by means of a non-elastic rope between the airplane and the piston. In such a case, the pressure acting against the piston is proportional to V^2 (velocity squared) of the piston.

If the airplane has a mass of 3000 kg, and a velocity of 48.9 m/s as it touches the ship. And if the airplane braking system is not used, but rather, a technique (of the three) starts to decelerate the airplane as soon as it touches the ship.

(1)
F=a(b)^c
Use this form to write expressions for the force supplied by the three techniques.
Knowing that F is the force, a and c are constants, and b is replaced by: velocity or acceleration or time or displacement.

(2)
For the first and second techniques, find the maximum acceleration experienced by the airplane and determine at what stage this acceleration is experienced.

(3)
If the braking system of the airplane is used, where it applies a total horizontal force of 6000 N to slow the airplane down, as soon as it touches the ship. Find the maximum acceleration experienced by the airplane when it is landing, if the third technique is used.

I have attempted to solve the problem as follow:

For the first branch:

Tech. 1 : F=ma so b=a (deceleration) and a=mass=3000 kg and c=1
Now, to determine the deceleration value, we use : V^2=(Vi)^2 - 2 (a) x which gives us a=17.07 ms^-2
Conclusion F=(3000)(17.07) N And here comes the confusion, since the force is constant, so how would it be written in the given form !?

Tech. 2 : F= kx so b=x (displacement) and a=k and c=1
Now, to determine k , we use 0.5mv^2=0.5kx^2 which leads to k= 1464 N/m
Conclusion, F= 1464x

Tech. 3 : Given that P ∝ v^2 ==> F/A ∝ v^2 ==> F= (cA).v^2 where cA is a constant, let it be K
Conclusion, F= K.v^2 where a=K and b=v and c=2

The second branch : the occurrence of the maximum acceleration can be found from the velocity-time graph, which shows that the max. acceleration occurs, for the first tech. at the beginning, and for the second tech. at the end.
the value of each is :

For tech. 2 : we have F=ma = 3000a and F=kx=(1464)(70)=102480 N

equating both, 3000(a)= 102480 N ==> a= 34.16 ms^-2

For Tech.1 I can not find it, since the force at the very beginning is highest, since the coefficient of static friction of the brake pads is used, which is larger than the kinetic. However, the effect of this static friction force of the braking system is instantaneous i.e at the first instant only. So, if I neglect this effect, the acceleration is constant, which is a=17.07 ms^-2 (found above). But I believe it shouldn't be like this !

For the third branch: we have F=K.v^2 , and since the braking system applies a force of 6000 N, then,
6000+F=K.v^2 , but here I have two unknowns; F and K.
Again, from the velocity-time graph I can see that the acceleration is maximum at the beginning, but I am unable to work it out, (I tried, but got wrong answers)

I think the thing you're missing in all these problems is that you're given x, the distance of the runway is 70m.  From that you can find either the accelerations (using exactly the equation you had above), but they're minimal accelerations and not actual magnitudes.  It doesn't give you the forces applied to the brake pads, for example.  You can use the brake force applied to figure out that force, though.  You also know that the energy stored in the elastic cable is equal to the kinetic energy of the plane as it lands, so that will give you its spring coefficient.  You also have no idea what the coefficient of proportionality for the water pump is, but you have to assume that it can't stop a plane because as the velocity drops it stops providing brake force.  6000 N they mention for the braking system also won't stop the plane in 70m, so I'd just discard that number as a horrible idea to start with.

You should have enough to help out now.  I don't just answer homework questions, but I provide advice.

Physics

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#### Dr. Stephen O. Nelson

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