Method 1: Impulse Method
This is the east method mathematically.
#I = Delta p#
-- Impulse causes momentum to change --
Where #I =# Impulse #= barFDeltat#
#p= momentum = mv #
#Delta p# = Change of momentum# = mv_f-mv_i#
#m# = mass of the bullet #= 5.5 xx10^-3 kg#
#v_f #= final velocity #= 0#
#v_i #= initial velocity# = 325 m/s#
#rArr barFDeltat= 0 - mv_i = mv_i#
#Deltat= (mv_i)/barF#
Method 2. Work-energy theorem
#W_"net" = DeltaK rArr vecF_"net"*vecd = 1/2mv_f^2 - 1/2mv_i^2 #
--Work causes kinetic energy to change --
where
d = stopping distance
F = frictional force
# - Fd = 0 - 1/2mv^2#
#d = 1/(2F)(mv_i^2)#
#Deltat = d/barv =(mv_i^2)/(2Fbarv) #
Because #barv = ½(v_i + v_f) = ½v_i#
#Deltat = (mv_i)/(F) #
Method 3 Kinematics and Newton's 2nd Law
#a = F/m #
#v_f^2 = v_i^2 - 2ad#
# rArr 0 =v_i^2 - 2(F/m)d #
#d =(mv_i^2)/(2F) #
#Deltat = d/barv =(mv_i^2)/(2Fbarv) = (mv_i^cancel(2))/(cancel(2)F*cancel(v_i)/cancel(2)) #
#Deltat = (mv_i)/F #
