Force and motion in Newton's laws and free body diagram

[theunloved]

A 40 kg girl and an 8.4 kg sled are on the frictionless ice of a frozen lake, 15 m apart but connected by a roped of negligible mass. The girl exerts a horizontal 5.2N force on the rope. What are the acceleration magnitudes of
(a) the sled
(b) the girl
(c) how far from the girl's initial position do they meet ?

Take positive x+ from left to right
From the free body diagram
On the sled, there are 3 forces in which the normal force and gravity force are equal and canceled out, so the only left is F, exerted from the girl
On the girl, normal and gravity are canceled out, and F is exerted back from the sled ( F is negative since it's opposite from the direction)
Sled : F app = m1 * a1 ----------> a1 = 5.2 / 8.4 = 0.61 m/s^2
Girl -F app = m2 * a2 -------------> a2 = -5.2 / 40 = -0.13 m/s^2


Distance traveled by the sled: x1 = 1/2 a1 * t^2 (1)
Distance traveled by the girl x2 = 1/2 a2 * t^2 (2)

x1 + x2 = 15
Plug (1) and (2) in that equation and find t
However, this is how I got confused, since for the equation (2), a is found negative, so I put in negative value, and ended up different answer from my friend, he claims that a should be positive because we should use its magnitude...So who is correct and why ? Can you tell me please ?

 

[Dick]

Your friend is right. The way you are doing this, you are just trying to find distance traveled by each one adding up to 15. You already know they are traveling towards each other. You don't need an explicit sign.