80 MOTION IN TERMS OF ACCELERATION. [CHAP. IV. 



98. A particle describes a circle as a central orbit about a point 0. 

 Prove that the sum of the velocities at any two points collinear with is 

 constant. 



99. A circle is described as a central orbit about a point on the circum 

 ference ; if the tangent to the circle meets the diameter through produced 

 in R, prove that the velocity of R is proportional to 



where a is the radius of the circle. 



100. A particle is projected from A with velocity &amp;lt;J(^}/OA 2 and moves 

 with an acceleration /x/(distance) 5 directed to 0, the direction of projection 

 making an angle a with OA. Prove that the particle will arrive at after a 

 time 



OA 3 a -sin a cos a 



101. A particle describes a circle as a central orbit about an excentric 

 point. In any diameter AB of the circle points S, S are taken such that 

 SA : S A=SJ3 : S J3=e. Prove that, if V and V are the velocities of the 

 particle at any point on the portion of the circle concave to iS&quot;, when the circle 

 is described about S and S respectively, and if V= V at A, then l/V-e/V 

 is constant. 



102. Prove that the acceleration with which a particle P can describe a 

 circle as a central orbit about a point S is inversely proportional to SP 2 .PP 3 , 

 where PP is the chord through S. 



If points are taken on the orbit such that the squares of their distances 

 from S are in arithmetic progression, the corresponding velocities are in 

 harmonic progression. 



103. Prove that the accelerations with which the same circle can be 

 described as a central orbit about two points R, S in its plane in the same 

 periodic time are in the ratio SGf 3 : RP 2 . SP, P being any point on the circle 

 and SG being a straight line drawn from S parallel to RP to meet the tangent 

 at P in G. 



104. A particle is moving with uniform velocity \/(^)/c 2 in a given 

 straight line, and when it is at a certain point it begins to have an acceleration 

 fj.r/(r 2 + b 2 ) 3 towards a point S distant a from the line. Prove that, if 

 c 2 &amp;gt;( 2 -f-6 2 ), there are two positions of the point for which the subsequent 

 orbit is a circle, and that the two circles cut at an angle &amp;lt;&amp;gt; given by 



c 2 sin a) 



105. A particle describes an ellipse of latus rectum 21 about the point X 

 where the axis meets a directrix. Prove that the acceleration is k*XP/(lSM 3 ), 

 where S is the focus corresponding to X, and M is the foot of the perpendicular 

 from P on the major axis. 



