An old 3000 -lb cannon fires an 18 -lb shell with an initial...

An old 3000 -lb cannon fires an 18 -lb shell with an initial velocity of $1500 \mathrm{ft} / \mathrm{s}$ at an angle of $30^{\circ}$ (you may assume this is the absolute velocity of the shell). The cannon rests on a horizontal surface and is free to move horizontally. Assuming that the barrel of the cannon is rigidly attached to the frame (no recoil mechanism) and that the shell leaves the barrel $5 \mathrm{~ms}$ after firing, determine $(a)$ the recoil velocity of the cannon, ( $b$ ) the resultant of the vertical impulsive forces exerted by the ground on the cannon.

Key Concepts

Conservation of Linear Momentum

This principle states that in the absence of external forces, the total momentum of a system remains constant. In the context of projectile motion and recoil, when a projectile is fired, the forward momentum of the projectile must be balanced by an equal and opposite momentum imparted to the launcher. This conservation is key in solving for the recoil velocity of systems such as cannons or firearms, where momentum is shared between the moving projectile and the recoiling body.

Impulse-Momentum Theorem

The impulse-momentum theorem describes the relationship between the impulse applied to an object and its change in momentum. Impulse, calculated as the product of force and the time interval over which it acts, results in a change in velocity or momentum of the object. This concept is crucial in impulsive events, such as the sudden force exerted by a gunpowder explosion in a cannon, and in determining both the recoil velocity and the reaction forces from the ground during rapid acceleration events.

Recoil Motion

Recoil motion refers to the backward movement experienced by a device when it ejects a projectile. It is a direct consequence of the conservation of momentum and is observed in systems where mass is expelled in one direction, leading to a reactive motion in the opposite direction. Analysis of recoil involves computing the momentum of the projectile and equating it with the momentum of the recoiling object, taking into account the mass and velocity changes involved.

Ground Reaction Forces

Ground reaction forces are the forces exerted by the ground on a body to support it and to respond to any vertical impulsive forces acting on the body. In scenarios involving rapid changes in momentum, such as the firing of a projectile from a cannon, the ground must exert a counteracting force to maintain equilibrium and balance the impulsive vertical forces. This concept is important in analyzing the overall force interactions and ensuring that all forces, including those from the supporting surface, are accounted for in the system's dynamics.

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