DYNAMICS OF A POINT. 291 



a direction making an angle cot" 1 2 with the distance ; deter- 

 mine the orbit, and prove that the time to the centre of force 



= A: 1S 2- 



1397. A particle is describing a circle under the action of a 

 iiit force in the centre, and the force is suddenly increased 



to ten times its former magnitude; prove that the next apsidal 

 distance will be equal to one fourth the radius of the circle. 



1398. A particle is describing a central orbit in such a 

 manner that the velocity at any point is to the velocity in a 

 circle at that distance as 1 : Jn', prove that p<xr* t p being the 

 perpendicular from the centre of force on the tangent at a point 



distance is r. Prove also that the force varies inversely 

 as r- +1 . 



1399 



. A particle acted on by a central force -^ f- -j 

 jected at a distance a, at an angle 45, and with vei 

 /Z -; determine the orbit and prove that the time from projec- 



1400. In an orbit described under a central force, the velo- 



)". the foot of the perjH-iidiiMilar from th- MQtN of 

 on the tangent, is constant: pro\- that tin- chord of cm \ 

 H through the- tvntre d :ant. 



rabola about a foi> 

 s to an apse, at whit -h joint tin- l.iw of force changes, 



crsely aa the distance till the purt'u-h 



to an a|--. v. ii.-n tlic former 1 1\\ is restored. No instunta- 

 changes being BUI-['O> 1. pTOff that tlie major axis of the 



-2 



