102
2 Particle Dynamics
At t = 0, r = R
∴ C 1 = R
∴ r = kt + R
(7)
Using (2) and (7) in (1)
d
dt
4
3
πρ(R + kt)
3 dx
dt
=
4
3
πρ(R + kt)
3 g
Integrating (R + kt)
3 dx
dt
=
(R + kt) 4 g
4k
+ C 2
But
dx
dt
= 0 when t = 0. C 2 = −
R 4
4k
g
The velocity after time t is therefore
v(t) =
g
4k
a + kt −
R 4
(R + kt) 3
2 Particle Dynamics
At t = 0, r = R
∴ C 1 = R
∴ r = kt + R
(7)
Using (2) and (7) in (1)
d
dt
4
3
πρ(R + kt)
3 dx
dt
=
4
3
πρ(R + kt)
3 g
Integrating (R + kt)
3 dx
dt
=
(R + kt) 4 g
4k
+ C 2
But
dx
dt
= 0 when t = 0. C 2 = −
R 4
4k
g
The velocity after time t is therefore
v(t) =
g
4k
a + kt −
R 4
(R + kt) 3