100
2 Particle Dynamics
2.65 Consider the equation for the variable mass
m
dv
dt
+ v
dm
dt
= F = 0
( 1 )
dm
dt
= k
(2)
∴ m
dv
dt
+ kv = 0
( 3 )
Integrating (2)
m =
dm =
kdt = kt + C 1
where C 1 = constant.
At t = 0, m = W
∴ m = kt + W
(4)
Using (4) in (3)
dv
v
= −
k
m
dt = −
kdt
kt + W
(5)
∴
dv
v
= −
kdt
kt + W
∴ ln v = − ln(kt + W ) + C 2
where C 2 = constant.
At t = 0, v = v 0 , C 2 = ln v 0 + ln W
∴ ln
v 0
v
= ln
1 +
kt
W
∴
v 0
v
= 1 +
kt
W
v =
ds
dt
=
v 0
1 +
kt
W
The distance travelled in time t
S =
s
0
ds = v 0
t
0
dt
1 +
kt
W
=
W v 0
k
ln
1 +
kt
W