The cars of a long coal train are filled by pulling them...

The cars of a long coal train are filled by pulling them under a hopper, from which coal falls into the cars at a rate of $10,000 \mathrm{~kg} / \mathrm{s}$. Ignoring friction due to the rails, what is the average force that the engine must exert on the coal train to keep it moving under the hopper at a speed of $0.50 \mathrm{~m} / \mathrm{s} ?$

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Key Concepts

Variable Mass Systems

In problems like these, where the mass of the system increases over time, the idea of a variable mass system is crucial. Analysis often involves incorporating the mass flow rate to understand how the system's momentum is altered. Recognizing that the system's momentum changes continuously due to new mass being added is fundamental in calculating the external force required to counterbalance the resulting momentum change.

Rate of Change of Momentum

Newton’s second law states that force is equal to the rate of change of momentum. In cases involving a continuous addition of mass, such as the coal falling into the moving train, the force needed can be determined by multiplying the mass flow rate by the constant velocity. This product gives the additional force necessary to maintain the train's constant speed.

Momentum and Its Conservation

Momentum, defined as the product of mass and velocity, is a central concept in dynamics. In this situation, as coal falls into the train, it contributes additional momentum. The principle of conservation of momentum helps in understanding how the momentum of the overall system changes when mass is added, thereby requiring an external force to maintain constant motion.

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