A metal cannonball of mass m rests next to a tree at the...

A metal cannonball of mass $m$ rests next to a tree at the very edge of a cliff $36.0 \mathrm{~m}$ above the surface of the ocean. In an effort to knock the cannonball off the cliff, some children tie one end of a rope around a stone of mass $80.0 \mathrm{~kg}$ and the other end to a tree limb just above the cannonball. They tighten the rope so that the stone just clears the ground and hangs next to the cannonball. The children manage to swing the stone back until it is held at rest $1.80 \mathrm{~m}$ above the ground. The children release the stone, which then swings down and makes a head-on, elastic collision with the cannonball, projecting it horizontally off the cliff. The cannonball lands in the ocean a horizontal distance $R$ away from its initial position. (a) Find the horizontal component $R$ of the cannonball's displacement as it depends on $m$. (b) What is the maximum possible value for $R$, and (c) to what value of $m$ does it correspond? (d) For the stone-cannonball-Earth system, is mechanical energy conserved throughout the process? Is this principle sufficient to solve the entire problem? Explain. (e) What if? Show that $R$ does not depend on the value of the gravitational acceleration. Is this result remarkable? State how one might make sense of it.

Key Concepts

Conservation of Mechanical Energy

This concept involves the principle that in a closed system with no non?conservative forces, the total mechanical energy (the sum of potential and kinetic energies) remains constant. In the problem, the gravitational potential energy of the raised stone is converted into kinetic energy as it swings down, and then further transforms during the collision with the cannonball.

Elastic Collisions

An elastic collision is one in which both momentum and kinetic energy are conserved. The head-on collision between the swinging stone and the stationary cannonball relies on this concept, necessitating calculations based on the simultaneous conservation of momentum and kinetic energy to find the postcollision velocity of the cannonball.

Projectile Motion

This is the study of an object that is launched into the air and moves under the influence of gravity alone, following a parabolic trajectory. In the problem, after the collision imparts a horizontal velocity to the cannonball, its subsequent motion off the cliff is analyzed using projectile motion principles, where the time of flight is determined by the vertical distance fallen under gravity.

Independence from Gravitational Acceleration

This concept highlights situations where gravitational acceleration cancels out of the final expressions for certain observable quantities. In the problem, despite gravity influencing the conversion of potential energy and the time of fall, the horizontal range turns out to be independent of the value of gravitational acceleration, a result that can be understood by noting how gravity simultaneously affects the fall time and the conversion of energy in the swing.

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