148 MOTIONS OF FREE PARTICLES. [CHAP. IX. 



165. Motion of a body under gravity. The frame of reference is 

 supposed to be fixed relatively to the Earth and to have its origin at a place 

 on the Earth's surface near which the motion takes place. Then it is 

 approximately true that the field of force is of intensity g directed vertically 

 downwards, and we treat this statement as exact. 



The equations of conservation of linear momentum show that the motion 

 takes place in the vertical plane through the direction of projection, and that 

 the horizontal velocity in that plane is constant. 



Let the axes of x and y be a fixed horizontal and vertical in this plane, the 

 positive sense of the axis y being upwards, then we have 



x = const. =u say. 



The kinetic energy of the body is \m 



The potential energy of the body in the field (Article 149) is mgy. 



Hence the equation of energy is 



\in (x 2 +y 2 ) + mgy = const. 

 This equation may be written 



\mu* jl + ^ | + mgy = const., 

 or, by choice of the constant 



giving 



where X Q is a constant of integration. This equation represents a parabola 

 with axis vertical and vertex upwards, the point (# , y ) being the vertex. 



166. Motion under a central force. For a particle under the action of 

 a force to a fixed point which is a function of the distance from that point, 

 the principles of the conservation of energy and momentum supply all the 

 first integrals of the equations of motion. Drawing a plane through the fixed 

 point and the tangent line to the path at any instant, the linear momentum 

 perpendicular to this plane is constantly zero, so that the particle moves in 

 the plane. The moment of momentum about an axis through the fixed point ' 

 perpendicular to the plane remains constant, and this gives us an equation of 

 the form 



pv = h, 



where the notation is that of Article 50, and we see that mh is the moment of 

 momentum about this axis. The kinetic energy of the particle is -|mv 2 , and 

 the potential energy of the particle in the field is 



f 



mfdr, 



