When only the air of a sand-blasting gun is turned on, the...

When only the air of a sand-blasting gun is turned on, the force of the air on a flat surface normal to the stream and close to the nozzle is $20 \mathrm{N}$. With the nozzle in the same position, the force increases to $30 \mathrm{N}$ when sand is admitted to the stream. If sand is being consumed at the rate of $4.5 \mathrm{kg} / \mathrm{min}$, calculate the velocity $v$ of the sand particles as they strike the surface.

Key Concepts

Conservation of Momentum

This principle states that the total momentum of a closed system remains constant in the absence of external forces. When dealing with collisions or streams of particles, the momentum lost by the moving particles is transferred to the object they impact, resulting in a force. This concept is fundamental in analyzing how momentum changes produce observable effects such as impact force.

Mass Flow Rate

Mass flow rate quantifies the amount of mass passing through a given point per unit time. It is essential in dynamics because it connects with the momentum carried by a stream of particles or fluid. By combining the mass flow rate with particle velocity, one can determine the rate of momentum transfer, which is directly related to the force exerted on an impacted surface.

Momentum Flux

Momentum flux refers to the rate at which momentum is transported per unit area by a moving fluid or stream of particles. In mechanics, it is used to calculate the force exerted by particles striking a surface, by relating how much momentum is transferred in a given time. This concept bridges the gap between the microscopic behavior of particles and the macroscopic force observed.

Impulse and Force Relationship

Impulse is defined as the change in momentum resulting from a force acting over a period of time. The relationship between impulse and force is given by the impulse-momentum theorem, which states that the force is equal to the rate of change of momentum. This framework is key in quantifying how continuous impacts or collisions can yield a steady force on an object.

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