IMPULSIVE FORCES 233 



As an illustration, consider the problem of firing a shot on board a 

 moving ship. The mass of the shot being small compared with that of the 

 ship, we may suppose the center of gravity of shot and ship to have 

 exactly the motion of the ship. The velocity of the shot relative to this 

 center of gravity may accordingly be taken simply to be that relative to 

 the deck of the ship. The work done by the powder in ejecting the shot 

 from the barrel is the same as though the ship were at rest, so that the 

 velocity of the shot relative to the ship will be the same as though the 

 ship were at rest. 



EXAMPLES 



1. A cart is moving with velocity V and a man on the cart throws out sand 

 horizontally from the back of the cart at the rate of m pounds per minute, the 

 sand having a velocity v relative to the road. At what rate is the man working ? 



2. A gun capable of firing a shot vertically upwards to a height h is placed 

 on an armored train running with velocity V. What is the greatest range to 

 which a shot can reach (a) behind the train, (6) in front of the train ? 



3. In the last question find the nearest point to the track, which is out of 

 range of the gun. 



4. A shell of mass M is moving with velocity V. An internal explosion gen- 

 erates an amount E of energy, and thereby breaks the shell into masses of which 

 one is k times as great as the other. Show that if the fragments continue to 

 move in the original line of motion of the shell, their velocities will be 



V + V2 kE/ M , V - V2 E/kM . 



5. Two men, each of mass M, stand on two inelastic platforms each of mass 

 m, hanging over a smooth pulley. One of the men, leaping from the ground, 

 could raise his center of gravity through a height h. Show that if he leaps with 

 the same energy from the platform, his center of gravity will rise a height 



2(M+m) 



IMPULSIVE FORCES 



189. There are many instances in dynamical problems in which 

 the action of a force begins and terminates within so short an 

 interval of time that the action may be regarded as instantaneous. 

 Such forces are called impulsive forces. As instances of impulsive 

 forces we may take the forces brought into play by the jerking of 

 an inextensible string, or by the collision between two hard bodies. 



The change of momentum produced by the action of an impul- 

 sive force is, in general, of finite amount. As the force only acts for 



