A coal car with an empty mass of 25 Mg is moving freely with a speed of 1.2 m / s under a hopper

step 1 So here on this problem we are given this car that has a mass of 25 megagram and is moving at a

A coal car with an empty mass of 25 Mg is moving freely with...

A coal car with an empty mass of $25 \mathrm{Mg}$ is moving freely with a speed of $1.2 \mathrm{m} / \mathrm{s}$ under a hopper which opens and releases coal into the moving car at the constant rate of $4 \mathrm{Mg}$ per second. Determine the distance $x$ moved by the car during the time that $32 \mathrm{Mg}$ of coal are deposited in the car. Neglect any frictional resistance to rolling along the horizontal track.

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Key Concepts

Mass Flow Rate

Mass flow rate is the measure of the amount of mass being added to or removed from a system per unit time. This concept is key when the rate of mass addition is constant, as it directly affects the system’s overall mass and, through momentum conservation, its velocity.

Integration of Differential Equations

When dealing with variable mass systems, one often derives differential equations that link velocity, mass, and time. Integrating these equations is essential for determining other dynamic quantities such as displacement, as the system evolves continuously over time.

Conservation of Momentum

In the absence of external forces, the total momentum of a system remains constant. This principle is crucial in situations where mass is added to or removed from a moving object, as it allows one to establish a relationship between the system’s instantaneous mass and its velocity.

Variable Mass Systems

A variable mass system is one in which the mass changes over time, often due to processes such as the addition or ejection of material. Analyzing such systems requires adapting standard dynamics equations, since the mass is not constant and directly influences the system’s acceleration and momentum.

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