We can write
dv
dt

as
dv
dt
 = dv
dx
 . dx
dt
 = dv
dx
 . v
.

In this example
- dv
dt
 = - dv
dx
 . v = c
x2
and hence
ó
õ 		-v dv = ó
õ 		c
x2
 dx

-c
x
 = -v2
2
 + k
and we are given c=4×105, and v=10 when x=104 so
k = -40 + 50 = 10 .
Hence when v=5 we have
-c
x
 = -12.5 + 10 = -2.5
and hence
x = 4×105
2.5
 = 160,000

. So the space craft moves at 5 km s-1 when it is 160,000 km from the Earth.