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Figure 1
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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.
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(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.
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Answer from Sia

Posted 4 months ago

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.

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