250 MOTION OF SYSTEMS OF PARTICLES 



8. Two billiard balls stand in contact, and a third ball is made to strike 

 them simultaneously, and is observed to remain at rest after the impact. 

 Show that e = f . 



9. A particle is projected from a point on a smooth horizontal plane, with 

 velocity V at an elevation a, and after striking the plane rebounds time after 



time. Show that its total time of flight is - , and that its total range 

 . F 2 sin2a g ^~ e ^ 



3 0(1- ) ' 



10. A player stands at a horizontal distance d from a wall, and throws a 

 ball towards the wall at an inclination a to the horizontal. Show that if it is 

 to return to him after bouncing, he must throw it with a velocity V given by 



2 e cos a (sin a /x, cos a) 



where e, /* are coefficients of elasticity and friction. 



11. In the last question consider the cases of (a) e = 0, (6) /i = tana, 

 (c) M > tan a. 



GENERAL EXAMPLES 



1. A particle is placed on the face of a smooth wedge which can slide 

 on a horizontal table ; find how the wedge must be moved in order that 

 the particle may neither ascend nor descend. Also find the pressure between 

 the particle and the wedge. 



2. It is required to run trains of 100 tons on a level electric railway, 

 with stations half a mile apart, at an average speed of 12 miles an hour, 

 including half a minute stop at each station. Prove that the electric loco- 

 motives must weigh at least an additional 8 tons, taking a coefficient of 

 friction of , and supposing the trains fitted with continuous brakes. 

 (Neglect passive resistances.) 



Prove that the railway can be worked by gravity, if the line is curved 

 downward between the stations to a radius of about 46,000 feet; and 

 that the dip between the stations will be about 20 feet, the inclines 

 at the stations about 1 in 33, and the maximum velocity about 23 miles 

 an hour. 



3. A cylinder of height h and diameter d stands on the floor of a rail- 

 way car, which suddenly begins to move with acceleration f. Show that 

 the cylinder will only remain at rest relative to the car if fis less than 

 both /j.g and dg/li. 



4. If a circular hoop is projected, spinning steadily without wobbling, 

 prove that the center describes a parabola, and that the tension of the rim 

 is the weight of a length v z /g of the rim, where v denotes the rim velocity 

 relative to the center of the hoop. 



