Answer
The final speed of the 50-g marble is 0.9 m/s
The final speed of the 20-g marble is 2.9 m/s
Work Step by Step
Let $m_A = 0.050~kg$ and let $m_B = 0.020~kg$.
Let $v_A$ and $v_B$ be the initial velocity of each marble.
Let $v_A'$ and $v_B'$ be the final velocity of each marble.
We can use conservation of momentum to set up an equation.
$m_A~v_A + 0 = m_A~v_A' + m_B~v_B'$
Since the collision is elastic, we can set up another equation.
$v_A - v_B = v_B' - v_A'$
$v_A - 0 = v_B' - v_A'$
$v_A' = v_B' - v_A$
We can use this expression for $v_A'$ in the first equation.
$m_A~v_A + 0 = m_A~v_B' - m_Av_A + m_B~v_B'$
$v_B' = \frac{2m_A~v_A}{m_A+m_B}$
$v_B' = \frac{(2)(0.050~kg)(2.0~m/s)}{(0.050~kg)+(0.020~kg)}$
$v_B' = 2.9~m/s$
We can use $v_B'$ to find $v_A'$.
$v_A' = v_B' - v_A$
$v_A' = 2.9~m/s - 2.0~m/s$
$v_A' = 0.9~m/s$
The final speed of the 50-g marble is 0.9 m/s
The final speed of the 20-g marble is 2.9 m/s
