GENERAL PRINCIPLES 33 



The greatest range or the maximum amplitude is when sin 2<x 

 = 1, or 2a = 90, and a = 45. It is necessary, therefore, to 

 project at an angle of less than 45. In the general equation of 

 Z, given x (the point it is desired the projectile should reach) 

 and the oblique speed v, the angle can be calculated at which 

 the projectile must be projected. 



It is not necessary here to. follow further the question of pro- 

 jectiles, but it may be noted that the trajectory is modified by 

 the resistance of the air, and that this diminishes the range (see 

 also 263). 



25. Central Force. Gravity is an example of what are called 

 central forces, because the earth draws bodies as though to a 

 central point, and in turn is drawn by the sun, the centre of our 

 planetary system. The direction of a central force always passes 

 through the centre, but the force can be attractive or repellant, 

 and act according to the distance of the moving body or the square 

 of that distance. Universal gravity causes a movement of the 

 stars in ellipses, but within the limits under study, that attraction 

 is perceptibly vertical to the earth and is apparently a parabola. 



Conversely, the movement or the trajectory being known, the 

 force can be determined. Take the case of a moving body con- 

 strained to describe a circle with a centre O (fig. 58) and find the 

 central force. Suppose the angular speed co to be constant. Then, 

 the required force must be F = mf, and the acceleration / is 

 here constant. It must also be noticed that, in a varying 

 curvilinear movement, the acceleration has two components. In 

 fact, when the moving body goes from M to M', the speed changes 

 from v to v -\- dv, dv being the acceleration in an infinitely small 

 space of time, dt, and the angle MOM' will be da. Draw M'A 

 equal and parallel to v. The 



acceleration is shown by j- 



dt 



As AB is the resultant of the 

 rectangular vectors BC and AC, 



the acceleration - - is the sum of 

 dt 



a tangential acceleration , 

 and of a centripetal acceleration "' 

 -jr* And there is no difficulty 



in seeing that the former is equal to - as has been already 



shown (2), whilst the latter is ,R being the radius of the 



R 



