An object can be projected upward at a specified velocity. If it is subject to linear drag, its altitude as a function of time can be computed as
z
=
z
0
+
m
c
(
v
0
+
m
g
c
)
(
1
−
e
−
(
c
/
m
)
t
)
−
m
g
c
t
where
z
=
altitude
(m) above the earth's surface
(
defined as
z
=
0
)
,
z
0
=
the
initial altitude (m),
m
=
mass
(kg),
c
=
a
linear drag coefficient (kg/s),
v
0
=
initial
velocity (m/s), and
t
=
time
(s). Note that for this formulation, positive velocity is considered to be in the upward direction. Given the following parameter values:
g
=
9.81
m/s
2
,
z
0
=
100
m,
v
0
=
55
m/s,
m
=
80
kg, and
c
=
15
kg/s, the equation can be used to calculate the jumper's altitude. Determine the time and altitude of the peak elevation (a) graphically, (b) analytically, and (c) with the golden-section search until the approximate error falls below
ε
s
=
1
%
with initial guesses of
t
l
=
0
and
t
u
=
10
s.