24 BEST AND MOTION 



5. A railroad runs due east and west in latitude X. At what rate 

 must a train travel along the road to keep the sun always directly 

 south of it? 



6. Determine the true course and velocity of a steamer going due north 

 by compass at 10 knots through a 4-knot current setting southeast; and 

 determine the alteration of direction by compass in order that the steamer 

 should make a true northerly course. 



7. A bicyclist rides faster than the velocity of the wind, and makes the 

 error of judging the direction of the wind to be the direction in which it 

 appears to meet him when he is in motion. Show that the wind will 

 always appear to be against him, in whatever direction he rides. 



8. One ship sailing east with a speed of 20 knots passes a light- 

 ship at 11 A.M. ; a second ship sailing south at the same rate passes the 

 same point at 1 P.M. At what time are they closest together, and what is 

 then the distance between them ? 



9. Two particles move with velocities v and 2 v respectively in oppo- 

 site directions, in the circumference o,f a circle. In what positions is their 

 relative velocity greatest and least, and what values has it then? 



10. Find the relative motion of two particles moving with the same 

 velocity v, one of which describes a circle of radius a while the other moves 

 along a diameter. 



11. Two particles move uniformly in straight lines. At a given time 

 the distance between them is a and their relative velocity is F, the com- 

 ponents of the latter in the direction of a and perpendicular to it being u 

 and v. Show that, when they are nearest together, their distance is av/V, 

 and that they arrive at this position after the interval au/ V 2 . 



12. Three horses in a field are at a certain moment at the vertices of 

 an equilateral triangle. Their motion relative to a person driving along a 

 road is in a direction round the sides of the triangle (in the same sense), 

 and in magnitude equal to the velocity of the carriage. Show that the 

 three horses are moving along concurrent lines. 



13. Two points describe concentric circles, of radii a and Z>, with speeds 

 varying inversely as the radii. Show that the relative velocity is paral- 

 lel to the line joining the points when the angle between the radii to 

 these points is 



cos 



-i 



2ab 



14. A stone dropped from a balloon moving horizontally is observed to 

 be 4 seconds in the air, and to strike the earth in a direction making an 

 angle of 15 with the vertical. Find the velocity of the balloon. 



