Question

equation for the system and solve for $v_{2f}$
$v_{2f}=\frac{m_1v_{1f}+m_2v_{2f}-m_1v_{1f}}{m_2}$
$=\frac{(1.60\\ kg)(4.00\\ m/s)+(2.10\\ kg)(-2.50\\ m/s)-(1.60\\ kg)(3.00\\ m/s)}{2.10\\ kg}$
$v_{2f}=-1.74\\ m/s$
(b) find the compression of the spring.
use energy conservation for the
$e_i = e_f$
system, noticing that potential energy
$\frac{1}{2}m_1v_{1f}^2+\frac{1}{2}m_2v_{2f}^2 + 0$
is stored in the spring when it is
$=\frac{1}{2}m_1v_{1f}^2+\frac{1}{2}m_2v_{2f}^2+\frac{1}{2}kx^2$
compressed a distance $x$.
substitute the given values and the
$x = 0.173\\ m$
result of part (a) into the preceding
expression, solving for $x$.
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remarks the initial velocity component of block 2 is $-2.50\\ m/s$ because the block is moving to the
left. the negative value for $v_{2f}$ means that block 2 is still moving to the left at the instant under
consideration.
question is it possible for both blocks to come to rest while the spring is being compressed? explain.
hint: look at the momentum in equation (1). (select all that apply.)
$square$ yes, if the two blocks initially have initial momenta with equal magnitudes and opposite directions.
$square$ yes, if the two blocks initially have equal masses.
$square$ no, it is not possible.
$square$ yes, if the ratio of the two initial speeds of the blocks equals the inverse ratio of their masses.
$square$ yes, if the two blocks initially have initial velocities with equal magnitudes and opposite directions.

Explanation:

According to the law of conservation of momentum \(p = m_1v_{1i}+m_2v_{2i}=m_1v_{1f}+m_2v_{2f}\). If both blocks come to rest (\(v_{1f} = v_{2f}=0\)), then \(m_1v_{1i}+m_2v_{2i}=0\), which can be rewritten as \(m_1v_{1i}=-m_2v_{2i}\). This means the initial momenta of the two - block system are equal in magnitude and opposite in direction. Also, \(m_1v_{1i}=-m_2v_{2i}\) can be expressed as \(\frac{v_{1i}}{v_{2i}}=-\frac{m_2}{m_1}\) (the ratio of the two initial speeds of the blocks equals the inverse ratio of their masses).

Answer:

Yes, if the two blocks initially have initial momenta with equal magnitudes and opposite directions.
Yes, if the ratio of the two initial speeds of the blocks equals the inverse ratio of their masses.