Two Carts on a Track
Students roll two carts toward each other on a low-friction track. Cart A has a mass of 2.0 kg and moves right at 3.0 m/s. Cart B has a mass of 3.0 kg and moves left at 2.0 m/s. They collide and stick together with magnets. After the collision, the combined carts move as one unit. A student claims that because Cart B is heavier, the combined carts must move to the left after the collision. Another student says the combined carts will be at rest because the speeds 'cancel out.' The class measures the combined velocity and finds it is not zero.
1. Use conservation of momentum to determine the velocity (magnitude and direction) of the combined carts after the collision.
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Take right as positive.
Initial momentum = m_A·v_A + m_B·v_B = (2.0 kg)(+3.0 m/s) + (3.0 kg)(−2.0 m/s) = +6.0 − 6.0 = 0 kg·m/s.
Total momentum before = 0, so total momentum after must be 0.
(2.0 + 3.0) kg × v = 0 → v = 0 m/s.
The combined carts are at rest after the collision. Note this contradicts the setup's claim that the measured velocity was not zero - so the measurement or the given values deserve scrutiny.
2. The measured combined velocity was not zero. Identify a likely source of the discrepancy and describe a test that would distinguish it from an error in the momentum reasoning.
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Possible sources: friction on the track (external force), the track not level (gravity component), carts not actually sticking, or measurement error in speed. The momentum reasoning itself is sound if no external net force acts. A distinguishing test: repeat with the track carefully leveled and measure speeds with a motion sensor at multiple points before and after collision. If the combined velocity shrinks toward zero as friction is reduced, friction/external forces explain the discrepancy; if it stays nonzero with negligible friction, the initial speed or mass values are wrong. This separates a model-assumption failure from an arithmetic error.
3. Explain why the 'heavier cart wins' claim is incomplete, using the momentum concept.
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Momentum depends on both mass and velocity (p = mv). Cart A's momentum is 6.0 kg·m/s right; Cart B's is 6.0 kg·m/s left. Equal and opposite momenta cancel regardless of which cart is heavier. The heavier cart does not 'win' unless its momentum is larger. The claim confuses mass with momentum.