Optimizing Airplane Manoeuvres: Calculating Time for a Circular Arc Turn

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Homework Statement

A pilot is flying his personal jet at 180m/s, approaching the airport. Near the airport the pilot sees the runway 300m horizontally to his left, but he is moving parralel to it. He banks left quickly, turning in level, circular arc, until the plane has moved 150 m, then banks quickly right, again turning in a circular arc, and ends up flying directly over the runway. How quickly does the pilot do this?

Homework Equations

The Attempt at a Solution

i have assumed that the flight is at the same altitude and speed during the manoevre.
would i use centripetal motion equations to solve this question because the plane moves in a circula arc, and would the distance from the runway to the plane be the radius?

 

[tiny-tim]

Hi Loading!

would i use centripetal motion equations to solve this question because the plane moves in a circula arc, and would the distance from the runway to the plane be the radius?

since the speed is constant, you only need to know the total distance

(and no, there are two arcs)

 

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but how would i find the distance in an arc? should i use the equation for circumfrence of a circle? or should i assume it as if it is a straight line?
Also after finding the distance of the first arc i can multiply it by two right ?

 

[tiny-tim]

but how would i find the distance in an arc? should i use the equation for circumfrence of a circle?

yes, but with θ instead of 2π

(it's a quarter-circle, so θ = π/2)

Also after finding the distance of the first arc i can multiply it by two right ?

the two arcs are identical, so yes

 

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i seem to be getting the wrong answer...
for circumfrence i am getting 471.23m
therefor the distance = 942.4m

and then to find time i used time=distance/speed and got 5.2s

my textbook has different answer.
Am i wrong or is there an error in the book?

 

[tiny-tim]

i seem to be getting the wrong answer...
for circumfrence i am getting 471.23m
therefor the distance = 942.4m

what formulas are you using?

 

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c=3.14/2(300)
circumfrence = pi divided by 2 multiplied by radius

and then i multiplied the circumfrence by 2

 

[tiny-tim]

no, circumference = pi times 2 multiplied by radius (2πr)

(and the radius isn't 300)

 

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yea but its 1/4 of a circle
and isn't 300 the randius because the distace from the runway to the plane is 300

 

[tiny-tim]

yea but its 1/4 of a circle

oh i see

but you used the word "circumference", and that means all the way round

anything else is just an "arc-length" (or an "arc")

… and isn't 300 the randius because the distace from the runway to the plane is 300

no, draw it carefully … there are two arcs

 

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yea but arnt both arc the same size?
also why isn't the radius 300?

 

[haruspex]

It isn't entirely clear, but it seems to me that you are supposed to assume the pilot is already level with the start of the runway when he realizes his error. I.e. it is directly to his left. (The 'plane has moved 150m' only makes sense, in relation to the total of 300m, if that is how much closer the plane has moved to the desired line of flight - it's not the length of the path travelled. That being so, if we don't know how far the pilot is short of being level with the runway then there's not enough information to solve the problem.)
This means two quarter circles won't do it either. Loading, try drawing the diagram on that basis.

 

[tiny-tim]

… This means two quarter circles won't do it either.

no, i don't agree …

… the pilot sees the runway 300m horizontally to his left, but he is moving parralel to it. He banks left quickly, turning in level, circular arc, until the plane has moved 150 m …

… it must mean 150 m left (ie halfway), so there are two quarter circles

 

[MrWarlock616]

tiny-tim, correct me if I'm wrong, the two arcs have equal radii of 150 m right? If yes, then kinematical equations will give us the time it took to cover an arc of π/2 radians, right?

 

[tiny-tim]

π/2 radians for each arc, yes

 

[haruspex]

… it must mean 150 m left (ie halfway), so there are two quarter circles

I agree with that, but two quarter circles will mean he has not only moved 300m to the left but also 300m further on. He will have overshot.

 

[tiny-tim]

how can he overshoot?

the runway is infinitely long! ​

 

[haruspex]

how can he overshoot?

the runway is infinitely long! ​

I don't see any such statement in the OP. If we don't have to worry aout overshooting then, as I previously posted, there's insufficient data to solve the problem. Why not fly in a very wide arc until halfway across to the runway then back the other way? E.g. a 300m radius arc for 60 degrees.

EDIT: Unless, of course, we're supposed to find the minimum time?

 

[tiny-tim]

but the question specifically says …

He banks left quickly … until the plane has moved 150 m, then banks quickly right … and ends up flying directly over the runway.

so the first bank moves him 150m closer, and the second bank moves him the other 150m closer (= 300m)

 

[haruspex]

but the question specifically says …


so the first bank moves him 150m closer, and the second bank moves him the other 150m closer (= 300m)

So? Pick any angle theta. Set a turning circle radius 150m/(1-cos(theta)) and bank left until you've turned an angle theta. You will be 150m closer to the line of the airstrip. Perform the same turn to the right. You are now in the line of the airstrip and flying along it.
To make progress we must either assume that the pilot must have no net forward motion from the manouevre, or that the time for the manouevre is to be minimised. There are hints supporting each of those in the OP.