.304 BOOK OF MATHEMATICAL PROBLEMS. 



resistance is m , and starts from the point where 8= ; 

 c 



prove that the velocity acquired in falling to the lowest point 

 is 



1452. A heavy particle slides on a smooth curve, whose plane 

 is vertical, in a medium whose resistance varies as the square of the 

 velocity, and in any time describes a space which is to the space 

 described in the same time by a particle falling freely in vacuo as 

 1 :2?i; prove that the curve is a cycloid, the vertex being the 

 highest point, and that the starting point of the particle divides 

 the arc between two cusps in the ratio 2n 1 : '2n + 1 . 



1453. A point describes a straight line under acceleration 

 tending to a fixed point and varying as the distance ; prove that 

 the corresponding point of the hodograph will move under the 

 same law of acceleration. 



1454. The curves 



r m = a m cos 7/i0, r n = a" cos nO 



will be each similar to the hodograph of the other when described 

 about a centre of force in the pole, provided that 



m n 

 Prove this property geometrically for both curves when m = 1 . 



1455. A point describes a curve in such a manner that its 

 hodograph is described as if under the action of a central force, 

 and T y N are the tangential and normal accelerations of the point; 

 prove that 



P 



p being the radius of curvature, s the arc measured from a fixed 

 point, and c a constant. 



