Figure 1
Figure 2
3. (12 points, sugzested time 25 minutes) Figure 1. Cat A is initially moving to the right, and Cart B is initially moving to the left. Cart A is in.iिistyg to fle Iight, as sinowinim Figure 2.
(a)Inficate winich rait, if either, has the greater mass.
Cant $A$ $\qquad$ Cart B $\qquad$ The carts have the same mass. $\qquad$
Brieffy justify your amswer ming qualitative reasoning beyond referencing equations.
6
t

$\mathbf{E}$

$=$
$=$
$=$
$=$
$=$
$=$
$=$
$=$
$=$
$=$
$=$
(b) Cant A hes meass $m_{A}$ and finitial speed $v_{A}$. Cart $B$ has mass $m_{B}$ and initial speed $v_{B}$. Starting winh conservation of momentum, derire an equation for the velocity $v_{f}$ of the two-cart system after the coltision Express your answerin terms of $m_{A}, m_{B}, v_{A}, v_{B}$, and physical constants as appropriate. Begin your derivaire by wrifing a İmatamemtal physics principle or an equation from the reference book.
(c) Indicate whefher fhe equation you derived in part (b) agrees with your reasoning in paut (a).

Explain your reasoning.

Question

Figure 1
Figure 2
3. (12 points, sugzested time 25 minutes) Figure 1. Cat A is initially moving to the right, and Cart B is initially moving to the left. Cart A is in.iिistyg to fle Iight, as sinowinim Figure 2.
(a)Inficate winich rait, if either, has the greater mass.
Cant $A$ $\qquad$ Cart B $\qquad$ The carts have the same mass. $\qquad$
Brieffy justify your amswer ming qualitative reasoning beyond referencing equations.
6
t

$\mathbf{E}$

$=$
$=$
$=$
$=$
$=$
$=$
$=$
$=$
$=$
$=$
$=$
(b) Cant A hes meass $m_{A}$ and finitial speed $v_{A}$. Cart $B$ has mass $m_{B}$ and initial speed $v_{B}$. Starting winh conservation of momentum, derire an equation for the velocity $v_{f}$ of the two-cart system after the coltision Express your answerin terms of $m_{A}, m_{B}, v_{A}, v_{B}$, and physical constants as appropriate. Begin your derivaire by wrifing a İmatamemtal physics principle or an equation from the reference book.
(c) Indicate whefher fhe equation you derived in part (b) agrees with your reasoning in paut (a).

Explain your reasoning.

Accepted Answer

∻Solution∻ ‖ a ⋮ To determine which cart has the greater mass, we can use qualitative reasoning based on the motion of the carts after the collision. Since Cart A is initially moving to the right and Cart B is moving to the left, and after the collision, the combined system moves to the right, it suggests that Cart A has a greater momentum. This implies that Cart A has a greater mass because momentum is the product of mass and velocity. ‖ ‖ b ⋮ To derive the equation for the final velocity $v_f$ of the two-cart system after the collision, we start with the conservation of momentum principle. The total momentum before the collision must equal the total momentum after the collision. Initial momentum: $m_A v_A + m_B (-v_B)$ Final momentum: $(m_A + m_B) v_f$ Setting the initial momentum equal to the final momentum: $$m_A v_A + m_B (-v_B) = (m_A + m_B) v_f$$ Solving for $v_f$: $$v_f = \frac{m_A v_A - m_B v_B}{m_A + m_B}$$ ‖ ‖ c ⋮ The equation derived in part (b) agrees with the reasoning in part (a). If Cart A has a greater mass, $m_A > m_B$, then the term $m_A v_A$ will dominate, resulting in a positive $v_f$, indicating motion to the right, which is consistent with the observation that the combined system moves to the right after the collision. ‖ ∻Answer∻ ⚹ Cart A has the greater mass. ⚹ ∻Key Concept∻ ⚹ Conservation of Momentum: The total momentum of a closed system remains constant if no external forces act on it. Equation: $m_A v_A + m_B (-v_B) = (m_A + m_B) v_f$ ⚹ ∻Explanation∻ ⚹ By analyzing the motion of the carts before and after the collision, we can infer that Cart A has a greater mass because the combined system moves in the direction of Cart A's initial motion. The derived equation confirms this reasoning. ⚹

Sia