The forces acting on the body are #mg# acting downwards and #kv# acting upwards.
Therefore,
According to Newton's Second Law
The acceleration is #a=(dv)/dt#
#m(dv)/dt=mg-kv#
Dividing by #m#
#(dv)/dt=g-k/mv#
Let #k/m=b#
Then,
#(dv)/dt=g-bv#
#(dv)/(g-bv)=dt#
Integrating both sides
#int(dv)/(g-bv)=int1*dt#
#-1/bln(|g-bv|)=t+C_1#
#ln(|g-bv|)=-bt-bc_1#
#g-bv=Ce^(-bt)#
Plugging in the initial conditions, #v=0# when #t=0#
#g-0=C*1#, #=>#, #C=g#
Therefore,
#bv=g-g e^(-bt)=g(1-e^(-bt))#
But #b=k/m#
#v=(mg)/k(1-e^(-kt/m))#
Plugging in the limiting speed,
#v_l=lim_(t->+oo)(mg)/k(1-e^(-kt/m))=(mg)/k#
But #v_l=160#
Therefore,
#(mg)/k=160#, #=>#, #m/k=160/g#
Therefore,
#v=160(1-e^(-g/160*t))#
When #t=5s#
The acceleration due to gravity is #g=32fts^-2#
#v(5)=160(1-e^(-32/160*5))=160(1-1/e)=101.1fts^-1#
The speed after #5s# is #=101.1fts^-1#
