Question

would be $f_{21}$. an easy way to remember this is from comes before on just like $f$ comes before $o$ alphabetically.

therefore, by newtons third law, $f_{21} = -f_{12}$ or $m_1a_1 = -m_2a_2$. replace the acceleration on both sides of the equation with the change in velocity over time. then, simplify so that all the terms with initial velocities are on the left and the final velocities are on the right.

what is the correct simplified equation?

$m_1v_{1,i} + m_2v_{2,i} = m_1v_{1,f} + $ ...
$m_1v_{1,i} + m_2v_{2,i} = m_1v_{1,f} - $ ...
$m_1v_{1,i} - m_2v_{2,i} = m_1v_{1,f} - $ ...
$m_1v_{1,i} - m_2v_{2,i} = m_1v_{1,f} + $ ...

Explanation:

Step1: Recall Newton's Third Law and Acceleration Definition

We start with \( m_1a_1=-m_2a_2 \). Acceleration \( a \) is defined as \( a = \frac{v_f - v_i}{t} \), where \( v_i \) is initial velocity, \( v_f \) is final velocity, and \( t \) is time. Substitute \( a_1=\frac{v_{1,f}-v_{1,i}}{t} \) and \( a_2=\frac{v_{2,f}-v_{2,i}}{t} \) into the equation:
\[ m_1\frac{v_{1,f}-v_{1,i}}{t}=-m_2\frac{v_{2,f}-v_{2,i}}{t} \]

Step2: Eliminate Time Variable

Multiply both sides of the equation by \( t \) to cancel out the denominator:
\[ m_1(v_{1,f}-v_{1,i})=-m_2(v_{2,f}-v_{2,i}) \]

Step3: Distribute the Mass Terms

Distribute \( m_1 \) and \( -m_2 \) on both sides:
\[ m_1v_{1,f}-m_1v_{1,i}=-m_2v_{2,f}+m_2v_{2,i} \]

Step4: Rearrange Terms for Initial and Final Velocities

Move all initial velocity terms to the left and final velocity terms to the right. Add \( m_1v_{1,i} \) and \( m_2v_{2,f} \) to both sides:
\[ m_1v_{1,i}+m_2v_{2,i}=m_1v_{1,f}+m_2v_{2,f} \]
(Assuming the options have a typo and the last term in the first option is \( + m_2v_{2,f} \), but from the given options, the first one \( m_1v_{1,i}+m_2v_{2,i}=m_1v_{1,f}+m_2v_{2,f} \) - matching the pattern of initial on left, final on right with correct signs from Newton's third law and momentum conservation derivation)

Answer:

\( m_1v_{1,i} + m_2v_{2,i} = m_1v_{1,f} + m_2v_{2,f} \) (Assuming the first option's missing term is \( + m_2v_{2,f} \), so the correct option is the first one with that correction, likely the intended equation for conservation of momentum derived from Newton's third law)