284 MOTION UNDEE A VARIABLE FOECE 



12. A particle is constrained to move in a circle of radius a, under an 

 attraction pr per unit mass to a point inside the circle distant c from its 

 center. If the particle be placed at its greatest distance from this point, 

 and started with an infinitesimal velocity, prove that it will pass over the 

 second quadrant of the circle in a time 



13. A particle describes an ellipse about a center of force in one focus. 

 Show that the velocity at the end of the minor axis is a mean proportional 

 between the velocities at the ends of any diameter. 



14. A comet describes a parabola. Show that its velocity perpendicular 

 to the axis of its orbit varies inversely as the radius vector from the sun. 



15. A comet of mass m, describing a parabola about the sun, collides 

 with an equal mass m at rest, and the masses move on together. Show that 

 their center of gravity will describe a circle about the sun as center. 



16. Assuming that a projectile, after allowing for variations in gravity, 

 describes an ellipse about the earth's center as focus, show that the maxi- 

 mum range on a horizontal plane through the point of projection, for a 

 given velocity v, is 



where R is the distance from the earth's center to the point of projection. 



17. When the earth is at the end of the major axis of its orbit, a small 

 meteor, of mass one rath of that of the sun, suddenly falls into the sun. 



2 



Show that the length of the year will be diminished by of itself. 



ra 



18. A planet P moving about the sun S picks up a small meteor, and 

 consequently has its velocity reduced by one nth of its former amount, 

 although unaltered in direction. Treating n as small, show that the eccen- 

 tricity of the planet's orbit will be reduced by 2 n(e + cos 0), where 6 is the 

 angle between SP and the major axis of the orbit. 



Show also that the new major axis will make an angle - with 

 the old axes. 



19. A particle describes an ellipse about the focus. Show that the 

 greatest and least angular velocities occur at the ends of the major axis, and 

 also that if a, j8 be these angular velocities, the mean angular velocity is 



c* + Vp 



