The 300 -kg and 400 -kg mine cars are rolling in opposite...

The 300 -kg and 400 -kg mine cars are rolling in opposite directions along the horizontal track with the respective speeds of $0.6 \mathrm{m} / \mathrm{s}$ and $0.3 \mathrm{m} / \mathrm{s}$. Upon impact the cars become coupled together, Just prior to impact, a 100 -kg boulder leaves the delivery chute with a velocity of $1.2 \mathrm{m} / \mathrm{s}$ in the direction shown and lands in the 300 -kg car. Calculate the velocity $v$ of the system after the boulder has come to rest relative to the car. Would the final velocity be the same if the cars were coupled before the boulder dropped?

Key Concepts

Conservation of Linear Momentum

This principle states that within an isolated system, the total linear momentum remains constant if no external forces are acting on it. In collisions and other interactions, even if objects combine or change their speeds, the sum of their momenta before and after the event must be equal. This concept is fundamental when analyzing collisions where multiple masses interact and become coupled or when additional masses join the system during motion.

Inelastic Collisions

An inelastic collision is one in which the colliding bodies stick together after impact, resulting in a shared velocity. Although kinetic energy is not conserved in such collisions, momentum is conserved. This concept is important when solving problems where objects couple together, as it allows for the calculation of the final velocity by setting the total pre-collision momentum equal to the total post-collision momentum.

System Mass Composition and Momentum Addition

In problems where additional masses, such as a projectile, join the moving system during or after a collision, it is essential to account for the momentum contribution of each mass. The combined system’s total momentum is the sum of the momentum of individual masses, computed using their masses and velocities. Correctly incorporating every component of the system is crucial for accurate analysis of the final motion.

Sequence of Collision Events

The order in which collisions occur can affect the intermediate momentum states of the system, even though momentum is ultimately conserved in the final state. When evaluating different scenarios, such as coupling vehicles before or after an additional mass is added, it is important to analyze each step separately to understand its effect on the overall system dynamics and to determine if the final velocity remains invariant despite the sequence of events.

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