IMPULSIVE FORCES 235 



Measure of an Impulse 



191. The change of momentum produced by an impulsive force 

 is called the impulse of the force. Thus if an impulse / acts on 

 a mass m, changing its velocity (or component of velocity in the 

 direction of the impulse) from u to v, we have 



I=m(v-u). (84) 



The force acting at any instant is, by the second law, equal 

 to the rate of change of momentum of the particle (or body) on 

 which it acts. If the force is of constant amount, the whole 

 change of momentum is equal to the product of the force by the 

 time over which it acts. If the force is of variable amount, the 

 change of momentum will be equal to the integral of the force 

 with respect to the time over which it acts. Thus if P is the 

 value of the force acting at any instant of the whole time t, we 

 see that the impulse 



= Pt, if the force is of constant amount, 



I- 



Pdt, if the force is of variable amount. 



Work done ly an Impulse 



192. The work done by an impulse / in changing the velocity 

 of a mass m from u to v will be 



the increase in the kinetic energy of the mass. Since / = m (v u), 

 we may write the expression for the work done in the form 



^ m (v u) (v + u) 

 + u\ 



Thus the work done ly an impulse is equal to the impulse multi- 

 plied by the mean of the initial and final velocities of the mass 

 acted upon. 



