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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 $(\mathrm{c})$ 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.

          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 $(\mathrm{c})$ 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.

Added by Marcus H.

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