Question

Two average forces. A steady stream of snowballs of mass m is shot perpendicularly into a wall at a speed of v. Each ball sticks to the wall. The figure gives the magnitude F of the force on the wall as a function of time t for two of the snowball impacts. Impacts occur with a repetition time interval t_r, last a duration time interval t_d, and produce isosceles triangles on the graph, with each impact reaching a force maximum F_max. During each impact, what are the magnitudes of (a) the impulse and (b) the average force on the wall? (c) During a time interval of many impacts, what is the magnitude of the average force on the wall? Express you answers in terms of t_r, t_d and F_max.

Answer 1

(a)   Impulse is nothing but area under the curve

    Hence the area of the triangle    is impluse    = (1/2) height * base

                 Impulse    = (1/2) Fmax (  t_d )

     (b)    The average   force is    = impulse / imact time ( t_d )  

                 Average force   = ( (1/2) Fmax (  t_d ) ) / ( t_d )    =  (1/2) Fmax

       Here impulse is  the result of (a) part

       (c) Let us take   10 impacts   ,    10 F(t) trangles

         And   the duration of time is  10 t_r   

           Hence   total area    = 10  (1/2) Fmax (  t_d )     =   5 Fmax (  t_d )  

             Average   force = 5Fmax (  t_d )      /   10 t_r       = (1/2)   Fmax (  t_d /  t_r   )

               For any number of hits we get the same answer