A solid block of water (m = 10 grams, Ti = -4 degrees Celsius) is shrink wrapped in an ultra thin super plastic (that has absolutely no effect on the interaction between its contents and the surrounding environment) and is placed as ammo inside a spring loaded cannon. The spring in these cannon has a “stiffness” of 250117 N/m and is compressed by 1 meter. The cannon is oriented so it fires horizontally.

Upon release, the spring expands accelerating the block from rest. The block scrapes against the inside of the cannon as it’s launched. After launch, the block bounces off the ground 8 times (leaving dents along the way) ultimately coming to a rest 20 meters from the front of the cannon. During the entire ordeal, 50% of the energy went into permanent deformation and 10% went to sound.
- If the final temperature of the block is 0 degrees Celsius, then by what percent has the block melted?
- otherwise, what is the final temperature of the block?

I have no idea how to begin this problem, any help is appreciated!

To solve this problem, we need to consider the conservation of energy and the thermal properties of the block. We'll break down the problem into several steps.

Step 1: Calculate the initial potential energy stored in the compressed spring:
The potential energy stored in a spring is given by the formula: PE = (1/2)kx^2, where k is the spring constant and x is the compression distance. In this case, k is given as 250117 N/m and x is 1 meter. So, the initial potential energy is:
PE = (1/2)(250117 N/m)(1 m)^2 = 125058.5 J

Step 2: Calculate the total energy input into the system:
From the problem statement, we know that 50% of the energy is lost as permanent deformation and 10% is lost as sound. Therefore, 40% of the energy is used to accelerate the block and overcome the friction with the cannon. So, the total energy input is:
Total Energy Input = PE / (1 - 0.4) = 125058.5 J / 0.6 = 208430.8 J

Step 3: Calculate the work done against friction:
The work done against friction is equal to the energy lost during the bouncing and denting process, which ultimately brings the block to a rest. This work can be calculated as:
Work against Friction = Total Energy Input - Final Kinetic Energy = 208430.8 J - 0 J (since the block comes to rest) = 208430.8 J

Step 4: Calculate the heat produced due to friction:
Since 40% of the energy went into permanent deformation and 10% went into sound, the remaining 50% is assumed to be converted into heat due to friction. So, the heat produced is:
Heat = Total Energy Input * 0.5 = 208430.8 J * 0.5 = 104215.4 J

Step 5: Calculate the amount of ice melted:
To calculate the amount of ice melted, we need to know the heat of fusion of ice, which is the amount of heat required to convert 1 gram of ice at 0 degrees Celsius to water at 0 degrees Celsius. The heat of fusion of ice is approximately 334 J/g. Since the block weighs 10 grams, the amount of ice melted is:
Amount of Ice Melted = Heat / Heat of Fusion = 104215.4 J / 334 J/g = 311.8 g

Step 6: Calculate the temperature change:
Finally, we can calculate the final temperature of the block using the specific heat capacity of ice. The specific heat capacity of ice is approximately 2.09 J/g°C. So, we can use the formula:
Heat = Mass * Specific Heat Capacity * Temperature Change
Rearranging, we get:
Temperature Change = Heat / (Mass * Specific Heat Capacity)
For the given values, the temperature change is:
Temperature Change = 104215.4 J / (10 g * 2.09 J/g°C) ≈ 4993.32 °C

Therefore, the final temperature of the block is approximately -4993.32 °C.