Russian problems Problem 82 The distance from A to B is d kilometers. A plane P is flying with constant speed, height and direction from A to B. Over a period of 1 second the angle PAB changes by a degrees and the angle PBA by b degrees. What is the minimal speed of the plane? Solution
: 20 p d Ã–(ab) kilometers per hour. Let the plane be at height h and a (horizontal) distance y from A. Let the angle PAB be q+a and the angle PBA be f . After 1 second, the angle PAB is q and the angle PBA is f+b . We have immediately that: h/y = tan( q+a ), h/(d-y) = tan f , h/(y+x) = tan q , h/(d-y-x) = tan( f+b ). Eliminating q , we obtain: h/y = (tan a + tan q )/(1 - tan a tan q ) = (a(y+x) + h)/(y+x-ah), where a = tan a . Hence x = a(h 2 + y 2 )/(h - ay). Similarly, eliminating f , we obtain x = b(h 2 + (d-y) 2 )/(h + (d-y)b). At this point I do not see how to make further progress without approximating. But approximating seems reasonable, since a and b , are certainly small, at least when expressed in radians. For example, typical values might be 10,000 ft for h and more than 10 miles for y or d-y and 500 mph for the aircraft speed. That gives x = 0.14 miles, so x/y = 0.014 and x/h = 0.07. So, let us neglect a/h, b/h, a/y etc. Then we get the simplified expressions: x = a(h 2 + y 2 )/h = b(h 2 + (d-y) 2 )/h. If a = b, then we quickly obtain y = d/2, h = d/2, x = ad. Assume a > b. Then we can solve for h, substitute back in and obtain an expression for x in terms of y. It is convenient to divide through by d and to write X = x/d, Y = y/d. Note that since we are assuming a > b, we require Y < 1/(1 + Ã– (a/b)). After some manipulation we obtain: X = ab(1 - 2Y)/ Ã– ((a-b)(b(1-Y) 2 - aY 2 ). Differentiating, we find that there is a minimum at Y = b/(a+b), which is in the allowed range, and that the minimum value of X is Ã– (ab). By symmetry, we obtain the same result for a < b and we notice that it is also true for a = b. So in all cases we have that the minimum value of x is d Ã– (ab). We are assuming a and b are small, so we may take a = a , b = b . However, the question specified that a and b were measured in degrees, so to obtain the final answer we must convert, giving: x = d ( p /180) Ã– (ab) , and hence speed = 20 p d Ã–(ab) kilometers per hour. Russian 81-90 (C) John Scholes jscholes@kalva.demon.co.uk 6 Nov 1998