24
1 Kinematics and Statics
x =
gt
k
−
g
k 2
1 − e
−kt
(4)
1.23 The equation of motion is
d 2 x
dt 2 = g − k
dx
dt
2
(1)
dv
dt
= g − kv
2
(2)
∴
1
k
dv
g
k
− v 2
= t + c
(3)
writing V 2 =
g
k
and integrating
ln
V + v
V − v
= 2kV (t + c)
(4)
If the body starts from rest, then c = 0 and
ln
V + v
V − v
= 2kV t =
2gt
V
∴ t =
V
2g
ln
V + v
V − v
(5)
which gives the time required for the particle to attain a velocity υ = 0. Now
V + v
V − v
= e
2kV t
∴
v
V
=
e 2kV t − 1
e 2kV t + 1
= tanh kV t
(6)
i.e.
v = V tanh
gt
V
(7)
The last equation gives the velocity υ after time t. From (7)
dx
dt
= V tanh
gt
V
x =
V 2
g
ln cosh
gt
V
(8)
x =
V 2
g
ln
e gt/v + e −gt/v
2
(9)