10 BEST AND MOTION 



lowest point of the wheel, and let X complete the parallelogram QPLX. 

 Obviously this parallelogram is similar to the parallelogram QTSH, corre- 

 sponding lines in the two parallelograms being at right angles. Thus 



QS : QT = QL : QP. 



So that on a scale in which the velocity of the carriage is represented in 

 magnitude by QP, the radius of the wheel, the velocity of the point Q will 

 be represented by QL. Thus the velocities of the different points on the 

 rim are proportional to their distances from L, their directions being in 

 each case perpendicular to the line joining the point to L. 



2. A battle ship is steaming at 18 knots, and its guns can fire projectiles 

 with velocities of 2000 feet per second relative to the ship. How must 

 the guns be pointed to hit an object the direction of 

 which from the ship is perpendicular to that of the 

 ship's motion? 



Let AB be the direction of the ship's motion, and 

 let us suppose the gun pointed in a direction AC. 

 Then the velocity of the shot relative to the ship 

 can be represented by a line Ap along A C, while that 

 of the ship relative to the sea can be represented by 

 a line Aq along AB. Completing the parallelogram 

 Aprq, we find that the diagonal Ar will represent the 



velocity of the shot relative to the sea in magnitude and direction. Hence 

 Ar must, from the data of the question, be at right angles to AB. If 6 is 

 the angle pAr through which the gun must be turned after sighting the 

 object to be hit, we have 



velocity of ship 

 - - * 



sin 6 = 



Ap velocity of firing of shot 



The velocity of the ship is 18 knots, or 18 nautical miles per hour. 

 Now 1 nautical mile = 1.1515 ordinary miles = 6080 feet, so that a 

 velocity of 18 knots is equal to 109,440 feet per hour, or 30.4 feet per 



30 4 

 second. Thus sin = -~- = .0152, whence we find that = 52' 16". 



Triangle of Velocities 



10. We can also compound velocities by a rule known as the 

 triangle of velocities. In fig. 3 the two velocities were represented 

 by Ap, Aq, and their resultant by Ar. The two velocities, how- 

 ever, might equally well have been represented by Ap, pr, and 

 their resultant by Ar, from which we obtain the following rule : 



