152 MOTIONS OF FREE PARTICLES. [CHAP. IX. 



2. Show that a gun at the sea level can command l/n 2 of the Earth's 

 surface if the greatest height to which it can send a shot is l/n of the Earth's 

 radius, variations of gravity due to height above the surface being taken into 

 account. 



3. A particle falls to the Earth from a height h. Prove that the time of 

 falling is (1 + fA/JB) ^(2%) approximately, where R is the Earth's radius, and 

 g is the acceleration due to gravity at its surface, and the square of h/R 

 is neglected. 



171. Motion in one plane, radial and transverse reso- 

 lutions. When the force acting on a particle is resolved into a 

 radial force R and a transverse force T always acting in one plane, 

 the equations of motion can be written 



m(r-r0 z )=R,\ 



By the process of Article 60 we can change the independent 

 variable from t to 0, and thus obtain the differential equation of 

 the path. 



Also when there is a work function W we have 



_ , m 



R= -^ , and T = - , 

 dr r W 



and then there is an energy equation of the form 

 i m (2 + ^202) -W + const. 



172. Examples. 



1. Writing h for r 2 0, and u for r~\ prove that the differential equation of 

 the path will be found by eliminating h between the equations 



, ^\_ 



" 1 / 



2. Supposing R=0, T=fj.r^ prove that the kinetic energy acquired in 

 describing a closed curve is 2/n x (area of curve). 



[This is an example of non-conservative positional forces referred to in 

 Article 152, it is in accordance with that Article to say that such forces 

 do not occur in natural systems.] 



173. Motion under several central forces. When a particle moves 

 under the action of central forces, directed to a number of different fixed 

 points, each force being a function of the distance from the point towards 

 which it acts, we cannot usually by help of general principles write down any 

 first integral of the equations of motion except the equation of energy. A 

 number of theorems relating to motions under such forces can however be 

 proved. We give a few examples with some indications of methods. 



