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



15. A particle moves in a straight line under a force tending to a fixed 

 point in the line which, at distance r, is equal to p/r 2 - &V(r 3 a), and starts 

 from rest at distance a +^(0?- 6 2 ). Prove that it will come to rest at distance 

 a-*J(a?-b 2 ) in time 7ra^/vV, and will oscillate between these distances. 



16. A particle moves along the axis #, starting from rest at x=a ; for 

 an interval ^ from the beginning of the motion the acceleration is - /*#, for a 

 subsequent interval t 2 the acceleration is /z#, and at the end of this interval 

 the particle is at the origin ; prove that 



^) tanh (vV 1 ^) = ! 



17. A particle moves with an acceleration always directed to a point 

 moving uniformly in a straight line, and the line joining the point to the 

 position of the particle at any time is normal to the path of the particle ; 

 prove that the path is an ellipse. 



18. A particle moves so that the angular velocity of the radius vector 

 from a fixed point and the acceleration along it are both constant, prove that 

 the acceleration at right angles to it varies as the sine of the angle between it 

 and a fixed straight line. 



19. A particle is moving in a parabola and at distance r from the focus 

 its velocity is v; show that its acceleration is compounded of ~r (v 2 r) 



parallel to the axis and - -=- ( ) along the radius vector outwards. 

 4 dr \r / 



20. A particle is describing an involute of a given curve ; prove that its 

 accelerations along the tangent and normal to its path are -r (sjs) and s\j/- 2 



respectively, where s is the arc of the given curve, ^ the angle which the 

 tangent makes with a fixed straight line. 



21. Prove that, if the acceleration of a point describing a tortuous curve 

 makes an angle ^ with the principal normal, then tan >// = - -5- , 



In the case of a plane curve the condition that the acceleration is always 

 directed to the same point is that the equation sin^+^; ^ J^ =0 must 



be satisfied at every point. 



22. The position of a point is given by the perpendiculars , rj on two 

 fixed lines containing an angle a with each other, prove that the component 

 velocities in the directions , rj are 



(+7)cosa)/sin 2 a and (77 + cos a)/sin 2 a. 



