167.] PLANE MOTION. 87 



body acquires the same velocity as in falling vertically through the 

 "height" of the plane. 



(3) Show that it takes a body twice as long to slide down a plane 

 of 30 inclination as it would take it to fall through the height of the 

 plane. 



(4) At what angle 6 should the rafters of a roof of given span 2 b be 

 inclined to make the water run off in the shortest time ? 



(5) Prove that the times of descending from rest down the chords 

 issuing from the highest (or lowest) point of a vertical circle are equal. 



(6) If any number of points starting at the same time from the same 

 point slide down different inclined planes, they will at any time / all be 

 situated on a sphere passing through the starting point and having a 

 diameter = 



(7) Show how to construct 'geometrically the line of quickest descent 

 from a given point : (a) to a given straight line, (b) to a given circle, 

 situated in the same vertical-;plaiie. 



(8) Analytically, the line &r quickest or slowest descent from a given 

 point to a curve in the sanje vertical plane is found by taking the 

 equation of the curve in polar co-ordinates, r=/(0), with the given 

 point as origin and the axis horizontal. The time of descending the 

 radius vector r is 't = ~^~2r/(g sin 6). Show that this becomes a maxi- 

 mum or minimum when tan 0=/(0)//'(0), according as /(0) +/"(#) 

 is negative or positive. 



(9) Show that the line of quickest descent to a parabola from its 

 focus, the axis of the parabola being horizontal, is inclined at an angle 

 of 60 to the horizon. 



167. Projectiles. With the same assumptions as in Art. 112, 

 the motion of a projectile reduces to that of a point subject to 

 the constant acceleration of gravity and starting with an initial 

 velocity V Q inclined to the horizon at an angle * different from 

 90. The angle e between the horizon and the initial velocity is 

 called the elevation of the projectile. 



