Question

cart 1 has an initial velocity and hits cart 2 which is stationary. after a perfectly inelastic collision, the combined carts are traveling with exactly one third the initial velocity of cart 1, and in the same direction. which is correct about the mass of cart 2?
the mass of cart 2 must be twice the mass of cart 1.
the mass of cart 2 must have decreased during the collision.
the mass of cart 2 must be one third the mass of cart 1.
the mass of cart 2 must be the same as the mass of cart 1.

Explanation:

Step1: Apply conservation of momentum

Let the mass of Cart 1 be $m_1$, its initial velocity be $v_1$, the mass of Cart 2 be $m_2$ (initially at rest, so its initial - velocity $v_2 = 0$). After the perfectly inelastic collision, the combined mass is $m_1 + m_2$ and the velocity is $\frac{1}{3}v_1$. According to the conservation - of - momentum formula $m_1v_1+m_2v_2=(m_1 + m_2)v_{final}$. Substituting the values, we get $m_1v_1+m_2\times0=(m_1 + m_2)\frac{1}{3}v_1$.

Step2: Simplify the equation

Since $m_1v_1=(m_1 + m_2)\frac{1}{3}v_1$ (and $v_1
eq0$), we can cancel out $v_1$ from both sides of the equation. We are left with $m_1=\frac{1}{3}(m_1 + m_2)$. Multiply both sides by 3 to get $3m_1=m_1 + m_2$.

Step3: Solve for $m_2$

Subtract $m_1$ from both sides of the equation $3m_1=m_1 + m_2$. We have $3m_1−m_1=m_2$, which simplifies to $m_2 = 2m_1$.

Answer:

The mass of Cart 2 must be twice the mass of Cart 1.