EXAMPLES 217 



the system is let go, W will remain at rest or move with uniform velocity, 

 provided -- 1 -- = and there is no friction anywhere. 



If this relation does not hold, find the acceleration of W. 



10. A particle, falling under gravity, describes 100 feet in a certain 

 second. How long will it take to describe the next 100 feet, the resistance 

 of the air being neglected ? 



If, owing to resistance, it takes .9 second, find the ratio of the resist- 

 ance (assumed to be constant) to the weight of the particle. 



sll. Prove that the line of quickest descent from any curve to any other 

 curve in the same vertical plane makes equal angles with the normals to 

 the two curves at the points at which it meets them. 



12, Find the position of a point on the circumference of a vertical 

 circle, in order that the time of rectilinear descent from it to the center 

 may be the same as that to the lowest point. 



13. Find the line of quickest descent from the focus to a parabola of 

 which the axis is vertical and vertex upwards, and show that its length is 

 equal to the latus rectum. 



^14. An ellipse is suspended with its major axis vertical. Find the diam- 

 etefdown which a particle can fall in the least time. What is the least value 

 of the eccentricity in order that this diameter may not be the major axis ? 

 15. A bullet is to be fired at a vertical target so as to hit it exactly at 

 right angles. If v is the velocity of the bullet and a the distance of the 

 target from the point of firing, show that the angle of elevation of the 



shot must be \ sin- 1 / ^M, and show that the point of the target which is 



hit will be at half the height of the point aimed at. 



16. A bullet is fired at a vertical target. Show that the projection of 

 the bullet on the target, as seen by the firer of the shot, appears to move 

 with uniform velocity. 



17. A gun fires two shots, one with velocity v at elevation a, and the 

 second with velocity v' at a smaller elevation a', in the same vertical plane. 

 Show that the shots will collide if the interval between firing is 



2 vv' sin (a a'} 

 g v cos a + v' cos a' 



18. A, B, C are three points in order in a horizontal line, AB being 

 640 feet ; a particle is projected from A with a velocity of 390 feet per 

 second in a direction making an angle tan~ 1 T 5 J with A C ; at the same instant 

 another particle is projected from B with a velocity of 250 feet per sec- 

 ond in a direction making an angle tan -1 f with BC ; show that these par- 

 ticles will collide, and find when and where. 



