Center of Mass and Motion of Objects: A Mechanics Exam Review Question

[jstep]

Homework Statement

I'm doing a review for a Mechanics exam I have tomorrow and this first problem given on the review struck me as odd.

Problem Statement:
An 89kg boat that is 6.6m in length is initially 7.1 m from the pier. A 27 kg child stands at the end of the boat closest to the pier. The child then notices a turtle on a rock at the far end of the boat and proceeds to walk to the far end of the boat to observe the turtle.

How far is the child from the pier when she reaches the far end of the boat? Assume there is no friction between boat and water.
Answer in units of m.

Homework Equations

The Attempt at a Solution

I tempted to just say 13.7m; but I'm guessing that somehow the child walking across the boat will cause it to slide across the water in the opposite direction. Is this somehow a momentum problem? But I have no way of knowing how quickly the child moves...

The other chain of thought i followed would cause the child's position to stay the same, while the boat moved. But I'm not sure how the different masses would play into this situation and surely it can't be that simple...

What am I missing here?

Thanks!

 

[gneill]

Conservation of momentum always applies, even without knowing the actual velocities involved. It leads to another useful property of isolated systems: Newton's first law applies to the center of mass of the system.

So, given that the boat and child are initially stationary, their center of mass is likewise stationary.

 

[jstep]

so the child's position would remain constant as the position of the boat changes?

 

[gneill]

No, the isolated system in this case comprises both the child and the boat.

 

[jstep]

so then the boat will remain stationary as the child walks across it?

v_{i, child} = 0
v_{i, boat} = 0

m_boatv_{i, boat} + m_childv_{i, child} = 0

v_{f, child} > 0
v_{f, boat} = ?

m_boatv_{i, boat} + m_childv_{i, child} = m_boatv_{f, boat} + m_childv_{f, child}

0 = 89v_{f, boat} + 29v_{f, child}

if v_{f, child} > 0

v_{f, boat} must be 0.

is this correct?

 

[gneill]

so then the boat will remain stationary as the child walks across it?

No. The boat and child must both move in such a way as to hold the center of mass stationary.

You won't have to consider the velocities at all, merely the location of the center of mass.