Conservation of momentum and astronaut

[ScullyX51]

Homework Statement

An astronaut is a distance L from her spaceship, and it is at rest with respect to the ship, when she discovers that her tether is broken. She tosses a wrench with a spees V_w in the direction opposite that of the ship in ordr to propel herself back to the ship. The astronaut has a mass M_A, and the wrench has a mass M_W.
1) What is the initial momentum (before toss) of the astronaut + wrench system? What is the final momentum?
2) Use conservation of momentum to solve for the final speed. V_A of the astronaut relative to the ship, in terms of M_A, M_W
, and V_w.
3) How long does it take her to reach the ship? Express your answer in terms of L, M_A, m_w, and V_W?
4) How far has the wrench traveled from its original position when the astronaut reaches the ship? Express your answer in terms of L, M_A, and m_W?

Homework Equations

conservation of momentum: P_i=P_F if F_net=0
x=x₀+v_x0+1/2axt²

The Attempt at a Solution

Please tell me if I've made any mistakes:
1) the initial momentum if zero since everything is at rest before she throws the wrench, and since it is an isolated system I have the following:
P_F=M_a(V_a)+ m_w(v_w)= P_i=0
2) final velocity is:
M_aV_a+m_wv_w=0
M^av_a=-m_wv_w
v_a=-m_wv_w/m_a
3)how long it takes her to reach her ship:
x=x₀+v_0x+1/2at²
(there is no acceleration due to gravity because they are in space)
0=L- (M_wV_w/m_a) t
L= (M_wV_w/m_a) t
I divided, inverted and multiplied, and came up with the following:
t= Lm_a/m_wv_w
4) I don't have any idea how to approach this part, and will appreciate any hints. Thank you.

 

[borgwal]

1,2,3 correct: 4: you know the speed of the wrench, and the time the astronaut travels.

Or, you know the ratio of their (constant) speeds, so the ratio of the distances they travel must be the same.

 

[djeitnstine]

You already solved for time T it took for the astronaut to reach the ship which will be the same time for the wrench. So simply use your momentum equation and solve for V of the wrench. and dx/dt = v of course.