31. We denote the magnitude of the frictional force αv, where α = 70 N
· s/m. We take the direction of the
boat’s motion to be positive. Newton’s second law gives
−αv = m
dv
dt
.
Thus,
v
v
0
dv
v
=
−
α
m
t
0
dt
where v
0
is the velocity at time zero and v is the velocity at time t. The integrals are evaluated with
the result
ln
v
v
0
=
−
αt
m
.
We take v = v
0
/2 and solve for time:
t =
m
α
ln 2 =
1000 kg
70 N
· s/m
ln 2 = 9.9 s .