EXAMPLES. 223 



147. A particle is projected vertically upwards from the surface of the 

 Earth with velocity %, and when its velocity is v and its height above the 

 surface is z the resistance is /tv 2 /(#4-2), where a is the Earth's radius. Prove 

 that, if z is always small compared with a, the velocity V with which it 

 returns to the point of projection is approximately given by the equation 



F 2 /w 2 = 1 - K v 2 lffa + (4/c 2 - AC) (v 2 /2#a) 2 , 

 variations of gravity with height being taken into account. 



148. A particle is projected vertically upwards in a medium in which the 

 resistance is kg (velocity) 2 . Prove that it returns to the point of projection 

 with kinetic energy diminished in the ratio 1 : 1+&F 2 , where Vis the velocity 

 of projection. 



Prove that in the same medium the angle 6 between the asymptotes of the 

 complete trajectory of a projectile is given by the equation 



U 2 /^ = cot 6 cosec d + sinh - 1 cot 0, 



where U is the terminal velocity and w the velocity when the projectile moves 

 horizontally. 



149. A particle moves under gravity in a medium whose resistance is 

 proportional to the velocity. Prove that the range on a horizontal plane is a 

 maximum, for given velocity of projection, when the angle of elevation at first 

 and the angle of descent at last are complementary. 



150. A particle is projected up a plane of inclination a under gravity and 

 a resistance proportional to the velocity. The direction of projection makes 

 an angle j8 with the vertical, the range R is a maximum and t is the time of 

 flight. Prove that, if U is the terminal velocity and V the velocity of 

 projection, then 



(i) 1 + ( VI U} sec ft = exp. (gt\ U\ 

 (ii) UV( U+ Fcos )/( F+ (7 cos ft) =g (R sin a + Ut\ 

 (iii) Z7F 2 sin/(F+ tfcosj8)=0cosa. 



151. A ,particle of unit mass describes a plane curve under a central 

 attraction equal to (p? + < 2 ) r when it is at a distance r from the origin, in a 

 medium whose resistance is 2< (velocity). Prove that its coordinates at 

 time t are 



e-** {XQ cos itt + p. ~ l (MQ + /c# ) sin /**}, 



e ~ Kt {2/o cos /** + P ~ l Oo + *#<>) sin /**} > 

 # , y being its initial coordinates and w , V Q its initial velocities. 



152. A particle moves under gravity in a medium whose resistance varies 

 as the square of the velocity, and u and v are the magnitudes of its velocity 

 at the two instants when its direction of motion makes an angle Jir with the 

 horizontal. Prove that, when it is moving horizontally its velocity is 



