Physics

Problem

Two low-friction carts, A and B, move on a straight horizontal track. Cart A has mass

m_A=0.50\,\text{kg}

and initially moves to the right at

v_0=2.4\,\text{m}\,\text{s}^{-1}

. Cart B has mass

m_B=1.0\,\text{kg}

and is initially at rest. The facing ends of the carts are covered with Velcro so that they stick together during the brief impact. Assume external horizontal forces are negligible during the collision. a. Derive an algebraic expression for the speed

v_f

of the combined carts immediately after impact in terms of

m_A

m_B

, and

v_0

. b. Calculate the numerical value of

v_f

for the given masses and initial speed. c. Determine the initial kinetic energy of the system, the kinetic energy immediately after the collision, and the percentage of the initial kinetic energy that is lost during the collision. d. The experiment is repeated with the same initial speed

v_0

, but the masses of the two carts are swapped (cart A now has

1.0\,\text{kg}

and cart B

0.50\,\text{kg}

). Without repeating detailed calculations, rank the two trials according to the percentage of kinetic energy lost (greatest to least) and justify your ranking qualitatively or quantitatively. e. Discuss two physical mechanisms by which the kinetic energy lost in these impacts could be transformed into other forms of energy.

Intuitive explanation

This is a perfectly inelastic collision — the key is that momentum is conserved even though kinetic energy is lost when the carts stick together. Since external horizontal forces are negligible, the total momentum before impact equals the total momentum after, letting you find the final velocity of the combined system. Think of it as redistributing the motion of the lighter, faster cart across the combined mass.

Intuitive explanation

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