THE INTERNAL WORK OF THE WIND. 



gains a finite velocity. In such a position, the air offers no resistance to its motion 

 and it falls in a straight line. When the velocity acquired is sufficiently great the 

 hand that guides the aeroplane can vary its angle of inclination so as to satisfy 

 both the condition a< 7° and the equation a = ^., . The aeroplane then follows a 

 trajectory which can be determined by the equations given above. This trajectory 

 inclines upward, until at a certain point B' the tangent is vertical. If now the 

 aeroplane, always under the control of the guiding hand, 1 is brought back to a ver- 

 tical position, the air offers no more resistance since the edge of the aeroplane is 

 presented. The aeroplane therefore ascends in a straight line. As the velocity at 

 B' is already known, the maximum altitude that the aeroplane can reach in a calm 

 atmosphere can easily be calculated. 



To sum up, the trajectory is made up of three parts (Fig. 5) : 

 from A to A' the aeroplane falls in a straight line and « = ; 

 from A' to B' the aeroplane follows the curve A'CB' and a = -^- > 

 from B' to B the aeroplane rises in a straight line aud « = 0. 



The problem is now to find the position of B with reference to A. 



1. From A to A' the aeroplane is propelled entirely by its weight ; we have 



therefore ~ = g, in which V= gt and S = j(/t~- By eliminating I we have also 



S =: -g- — . If we assume that the aeroplane falls in a straight line until it acquires 



a velocity of 10 meters per second, we find that AA' = 5.097465 m,, the velocity 

 at A = 0, and the velocity at A' = 10 meters per second. 



2. From A' to B' the trajectory is determined from the equations previously 

 established, by introducing in them the conditions actually prevailing at the begin- 

 ning of the curve A'B' ; these are V = 10, b = — * , and consequently cos b = 

 and sin b = 1. 



To find the velocity at C, the lowest point on the curve, let /*= in equation 

 (3); this gives, after numerical values have been substituted for all the constants, 

 V c = 22.1607 m. 



In the same way we find the velocity at B', in this case calling /3 = ^- which 

 gives Vb< = 8.31924 m. 



It has been remarked above that bv taking a as 7.808, the angle a will never 

 exceed 7 degrees ; in reality the maximum value of a corresponds to the minimum 

 of V, 8.319 m., and can be obtained from the formula a = ^ = '[,,,; which gives 

 a result of about 6° 30' for a, justifying the value arbitrarily assigned to c. 



1 The changing of the orientation of the a§roplane necessitates the expenditure of a certain 

 amount of energy which is nol taken accounl of here as it is very small. 



