Two astronauts, one of mass 60 kg and the other 90 kg, are initially at rest together in outer space. they then push each other apart. how far apart are they when the lighter astronaut has moved 15 m?

Let the mutual force between the two astronauts when they push apart be

F N, as shown in the figure below.

1st astronaut:

The mass is 60 kg.

Let the acceleration be a₁ m/s².

Then

F = 60a₁ (1)

2nd astronaut:

The mass is 90kg.

The acceleration, a₂, is given by

F = 90a₂ (2)

When lighter astronaut has traveled a distance of 15 m, the time taken, t, is given by

(1/2) a₁t² = 15

or

a₁ t² = 30 (3)

The distance traveled by the other astronaut is

d₂ = (1/2) a₂ t² (4)

Substitute (3) into (4)

d₂ = (1/2) a₂ (30/a₁) = 15 (a₂/a₁)

The separation between the astronauts is

D = 15 + d₂ = 15 + 15 (a₂/a₁) (5)

From (1) and (2), obtain

60a₁ = 90a₂

a₂/a₁ = 2/3 (6)

Substitute (6) into (5).

D = 15 + 15 (2/3) = 25 m

Answer: 25 m