Help challenging problem on energy and projectile motion

[jmb07]

One winter break, while visiting Scotland's Loch Ness, you stop to tour the ruins of Urquhart
Castle, which are found on the loch's western shore. A truck carrying barrels passes and, as it does, a
sealed barrel and a spare lid fall from the truck and land by the side of the road. The barrel comes to
rest at the very top of the snowy slope that leads down to the loch's shoreline. A few moments later, the
wind from the next car is just enough to push the barrel over the edge to slide, not roll, down the slick
slope. Once the barrel reaches the bottom, it begins to roll across the level shore. It rolls over exactly 3
times before rolling into a tree branch at the water's edge. Upon hitting the branch, the barrel is
knocked up and into the air, at what appears to be a 45 degree angle. It flies through the air for about 2
seconds before landing in the water. After it is in the water for a few moments, you notice that the
barrel is floating right side up with about half of it above the surface. To your astonishment, a small
aquatic creature, the likes of which you have never seen, begins to play with the barrel. Following one
particularly strong nudge to the barrel, which leaves the barrel bobbing up and down in the water, the
creature is scared away by shouts from other tourists. After recording that the barrel bobs up and down
once every 2.4 seconds you walk along the road to further investigate what you have just witnessed.
Your investigation reveals that the mass of the spare lid is 6.8 kg, the road is 14m above the shore, and
that distance from the bottom of the slope to the water's edge is 5.4m.
Now, you have enough information to answer a very important question, namely, was there any
whiskey in the barrel and, if so, how much?

 

[LowlyPion]

One winter break, while visiting Scotland's Loch Ness, you stop to tour the ruins of Urquhart
Castle, which are found on the loch's western shore. A truck carrying barrels passes and, as it does, a
sealed barrel and a spare lid fall from the truck and land by the side of the road. The barrel comes to
rest at the very top of the snowy slope that leads down to the loch's shoreline. A few moments later, the
wind from the next car is just enough to push the barrel over the edge to slide, not roll, down the slick
slope. Once the barrel reaches the bottom, it begins to roll across the level shore. It rolls over exactly 3
times before rolling into a tree branch at the water's edge. Upon hitting the branch, the barrel is
knocked up and into the air, at what appears to be a 45 degree angle. It flies through the air for about 2
seconds before landing in the water. After it is in the water for a few moments, you notice that the
barrel is floating right side up with about half of it above the surface. To your astonishment, a small
aquatic creature, the likes of which you have never seen, begins to play with the barrel. Following one
particularly strong nudge to the barrel, which leaves the barrel bobbing up and down in the water, the
creature is scared away by shouts from other tourists. After recording that the barrel bobs up and down
once every 2.4 seconds you walk along the road to further investigate what you have just witnessed.
Your investigation reveals that the mass of the spare lid is 6.8 kg, the road is 14m above the shore, and
that distance from the bottom of the slope to the water's edge is 5.4m.
Now, you have enough information to answer a very important question, namely, was there any
whiskey in the barrel and, if so, how much?

What is your approach to the problem?

 

[nlightner]

I would approach this problem by first analyzing the physical principles involved in the scenario. The key elements to consider are energy and projectile motion.

The barrel was initially at rest at the top of the slope, which means it had potential energy due to its position. When the wind pushed it over the edge, it gained kinetic energy as it slid down the slope. As it reached the bottom and began to roll, the kinetic energy was converted into rotational energy. This is why the barrel rolled over exactly 3 times before hitting the branch.

When the barrel hit the branch, it experienced a change in direction and its rotational energy was converted back into kinetic energy, causing it to be launched into the air at a 45 degree angle. This is an example of projectile motion, where an object is launched at an angle and follows a curved path due to the force of gravity.

To determine if there was any whiskey in the barrel, we need to calculate the initial kinetic energy of the barrel when it was launched into the air. This can be done using the mass of the spare lid (6.8 kg) and the distance it traveled from the bottom of the slope to the water's edge (5.4m). Using the equation for kinetic energy (KE=1/2mv^2), we can find that the initial kinetic energy of the barrel was approximately 207.36 Joules.

Next, we need to determine the potential energy of the barrel when it was at the top of the slope. This can be calculated using the mass of the barrel (unknown) and the height of the slope (14m). Since the barrel was at rest, the potential energy at the top of the slope would be equal to the kinetic energy when it was launched into the air. Therefore, we can set up an equation where PE=mgh=KE, and solve for the mass of the barrel.

Substituting the values, we get m=207.36/9.8/14=1.5 kg. This means that the barrel had a mass of 1.5 kg, which is significantly less than the mass of the spare lid. This indicates that there was no whiskey in the barrel, as it would have had a much greater mass if it were filled with liquid.

In conclusion, based on the information provided and the principles of energy and projectile motion, it is highly unlikely that there was any whiskey in the barrel. The small aquatic creature that was playing with