EXAMPLES. 69 



6. A point C describes a circle of radius r with angular velocity co about 

 the centre 0, and a point P moves so that CP is always equal to a and turns 

 with angular velocity o&amp;gt; in the plane of the circle described by C. Prove that 

 the angular velocity of OP is 



J {o&amp;gt; (R 2 + a 2 - r 2 ) + a&amp;gt; (7Z 2 - a 2 + r 2 )} /.ft 2 , 

 where .ft is the length of OP. 



7. Two points move uniformly in straight lines. At any time the 

 distance between them is a, V is their relative velocity, u and v are the 

 resolved parts of V parallel and perpendicular to the direction of a. Show 

 that, when they are nearest together, their distance is av/V t and that the 

 time until they arrive at this position is au/V 2 . 



8. Two points A and B move with uniform velocities u, v in two straight 

 lines containing an angle a ; prove that the time from the position in which 

 AB is least to that in which it is double its least value is 



JScv sin a/(u 2 + v 2 - 2uv cos a), 

 where c is the distance AB when A crosses the path of B. 



9. Prove that when a particle moves along a plane curve the velocity of 

 the foot of the perpendicular from the origin on the direction of motion is 

 rv/p, v being the velocity of the particle, r its distance from the origin, and p 

 the radius of curvature of its path. 



10. Two particles start simultaneously from the same point and move 

 along two straight lines, one with uniform velocity, the other with uniform 

 acceleration. Prove that the line joining the particles at any time touches a 

 fixed parabola. 



11. A particle moves with uniform acceleration along the tangent to its 

 path and describes arcs s l5 s 2 , s 3 in the %, ?i 2 , n s th seconds after any 

 particular instant ; prove that 



~ ^ = 0. 



12. Two boats start off to race with velocities v, v , and move with 

 accelerations /, / , the result being a dead heat. Prove that the length of the 

 course is 



13. A body is projected vertically upwards with velocity v, after a time t 

 a second body is projected vertically with velocity v (&amp;lt;v). If they meet as 

 soon as possible 



14. A particle moves in the axis x with acceleration p./x 2 towards the 

 origin, starting from rest at x=a. Show that the time of arriving at a 

 distance x is 



