EXAMPLES. 163 



8. A stream of particles originally moving in a straight line K with 

 velocity F is under the influence of a gravitating sphere of radius R, whose 

 centre moves with velocity v in a straight line intersecting the line K and 

 making with it an angle a. Prove that, if the distance of the sphere from the 

 line is originally very great, a length 



9 ft 



T 

 v sin a 



of the line of particles will fall upon the sphere, g being the force per unit 

 mass at the surface of the sphere. 



9. A particle is projected with velocity less than that from infinity under 

 a force tending to a fixed point and varying inversely as the nth power of the 

 distance. Prove that if n is not <3 the particle will ultimately fall into the 

 centre of force. 



10. A particle moves under a central force varying inversely as the nth 

 power of the distance (%>1), the velocity of projection is that due to a fall 

 from rest at infinity, and the direction of projection makes an angle /3 with the 



2 



radius vector of length R. Prove that the maximum distance is R cosec n ~ 3 /3 

 when n>3, and that the particle goes to infinity if n= or <3. 



11. Prove that the time of describing any part of a central orbit is 



rdr 



I, 



taken between appropriate limits, where F is the potential, and C and k are 

 constants depending on the initial conditions. 



12. Prove that, if a possible orbit under a central force cj> (r) is known, a 

 possible orbit under a central force <f> (r) + \r~ 3 can be found. In particular 

 prove that a particle projected from an apse at distance a with velocity 

 under an attraction 



will arrive at the centre in time 



a 



2 V /* \2-6// \n-Z 



13. A particle moves under a central force and is projected with velocity 

 v Q from a point at distance r Q in a direction making an angle a with the radius 

 vector. Prove that the apsidal distances are the real roots of the equation 

 forr 



where W is the work done by the central attractive force as the particle 

 moves from the point of projection to any point r, 6. 



112 



