The problem is as follows:
The figure from below shows two students whose masses are $m_1$ and $m_2$ ($m_1 < m_2$) are situated on both ends of a canoe situated in a lake with calm waters. Find the displacement that the canoe whose mass is $m_3$ experiences until the instant the students exchange their initial positions.
The alternatives given are as follows:
$\begin{array}{ll} 1.&\left(\frac{m_2 - m_1}{m_1 + m_2 + m_3}\right)L\\ 2.&\left(\frac{m_3 - m_1}{m_1 + m_2 + m_3}\right)L\\ 3.&\left(\frac{m_3 - m_2}{m_1 + m_2 + m_3}\right)L\\ 4.&\left(\frac{m_2}{m_1 + m_2 + m_3}\right)L\\ 5.&\left(\frac{m_3}{m_1 + m_2 + m_3}\right)L\\ \end{array}$
I'm confused exactly what sort of equation or analysis I can attempt to do in order to solve this question. I think that it is related with momentum and I can consider that when they exchange positions the momentum is preserved.
But I dont know exactly if should I say:
$p_1+p_2+p_3= p_{1f}+p_{2f}+p_{3f}$
Should this be the right way to do?. Can someone help me here please?.
Take the students and the boat to be one system. When the students swap their positions, all forces used are internal (assuming negligible resistance by water). There is no net external horizontal force, so the horizontal position of centre of mass of the system stays constant. Fix a stationary origin. Let the student with mass $m_1$ be $x_1$ units away, the one with mass $m_2$ be $x_2$ units away, and the the centre of mass of the boat of mass $m_3$ be $x_3$ units away from the origin. The abscissa of the centre of mass of the system is$$x=\frac{m_1x_1+m_2x_2+m_3x_3}{m_1+m_2+m_3}$$There is no change in the abscissa, in other words $\Delta x=0$.$$m_1\Delta x_1+m_2\Delta x_2+m_3\Delta x_3=0$$where $\Delta x_i$ denotes the displacement of $m_i$. Now, $m_1$ undergoes a displacement of $L$ to the right and $\Delta x_3$ to the right. $m_2$ undergoes a displacement of $L$ to the left and $\Delta x_3$ to the right. Thus$$\Delta x_1=\Delta x_3+L\\\Delta x_2=\Delta x_3-L$$Plug these into the earlier equation and solve to isolate $\Delta x_3$.