A rotary power plow is used to remove snow from a level...

A rotary power plow is used to remove snow from a level section of railroad track. The plow car is placed ahead of an engine that propels it at a constant speed of $20 \mathrm{km} / \mathrm{h}$. The plow car clears $160 \mathrm{Mg}$ of snow per minute, projecting it in the direction shown with a velocity of $12 \mathrm{m} / \mathrm{s}$ relative to the plow car. Neglecting friction, determine $(a)$ the force exerted by the engine on the plow car, $(b)$ the lateral force exerted by the track on the plow.

A rotary power plow is used to remove snow from a level section
of railroad track. The plow car is placed ahead of an engine that
propels it at a constant speed of $20 \mathrm{km} / \mathrm{h}$. The plow car clears $160 \mathrm{Mg}$ of snow per minute, projecting it in the direction shown with a velocity of $12 \mathrm{m} / \mathrm{s}$ relative to the plow car. Neglecting friction, determine $(a)$ the force exerted by the engine on the plow car, $(b)$ the lateral force exerted by the track on the plow.

Key Concepts

Conservation of Momentum

This principle asserts that in an isolated system, the total momentum remains constant if no external forces act. When mass is being ejected from a moving object, the momentum carried away by the mass requires a corresponding reaction force on the object. In problems where a machine expels material at a certain velocity relative to its frame, the conservation of momentum concept allows for computing the force exerted by or on the machine.

Rate of Change of Momentum

Force can be determined as the rate at which momentum changes with time. When a system expels mass continuously, the resulting force is calculated by multiplying the mass flow rate by the velocity at which the mass is accelerated relative to the inertial frame. This approach is central to understanding propulsion and reactive forces in many engineering and physics contexts.

Relative Velocity and Vector Analysis

In scenarios where an object ejects mass, the velocity of the ejected mass is often given relative to the moving object. To analyze the forces correctly, it is necessary to transform this relative velocity into the inertial frame of reference. This involves vector decomposition and understanding how the ejected mass's velocity components contribute to the overall momentum and force in different directions.

Force Decomposition

When forces act in multiple dimensions, especially when the ejected material has components in directions other than that of the object’s motion, it becomes important to resolve the forces into their components. This allows one to separately analyze the effects along the direction of motion (resulting in propulsion or resistance) and perpendicular to it (leading to lateral forces), ensuring a complete understanding of the dynamics at play.

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