First I shall prove the formula:

a = v dv dx     (1)

I shall write the formula for acceleration

dv dt

as

a = dv dt = dv dx . dx dt = v. dv dx .

From the problem, I know that a = -c/x², as vectors a and x have opposite signs. Using this and relation (1), I obtain: In this example

dv dt = v. dv dx = - c x2

and hence

ó õ v dv = ó õ -c x2 dx

v2 2 = c x + 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

12.5 = c x + 10

which gives

x =

4×105 2.5

= 160,000

. So the space craft moves at 5 km per second when it is 160,000 km from the Earth.

I solved the problem in two ways, as above using the relation just proved, and using conservation of energy. The methods are essentially the same and the constant given as c combines the gravitational constant G and the mass of the Earth.