T ^8 ANSWERS. 



(5) Compare the height of ascent in Ex. (4) to the distance fallen 

 through as obtained in (27), Art. 126. If v l be the velocity with which 

 the particle returns to the starting point, we find 



(6) v = v e- kt , 



(7) ^ = (i- 



Page 69. 



(i) w = TT radians ; v = 18.8 ft. per second. 



W (*)3i; (J)3- 



(3) -0.157- 



(4) 5- 



(5) (a) 402.1 ; (b) 25.1 seconds. 



Page 73. 



(1) r = v$t, 6 = <o/; hence r = v^B/a, a spiral of Archimedes. 



(2) About the pole O describe a circle of radius a and find its 

 intersection Q with the perpendicular to the radius vector OP drawn 

 through O ; then QP is the normal. Proof by Ex. ( i ) . 



(3) For the direction of v see Art. 31, Ex. (2). Resolving 

 v into v parallel to the track and v l along the tangent to the wheel, 

 it appears that v bisects the angle between these components ; hence 

 v = 2 v cos CAP, where C is the centre of the wheel, and A its lowest 

 point. 



(5) For the ellipse, r + r*= const. ; hence = , />. the pro- 



at at 



jections of the velocity on the radii vectores are equal. 



(6) The projections of the velocity on the radius vector and on 

 the focal axis are in the constant ratio e of the focal radius vector 

 to the distance to the directrix. It follows that the tangent intersects 

 the directrix in the same point as does the perpendicular to the radius 

 vector through O. 



(7) 40. 



