2. 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 can break it down into several steps:
Step 1: Calculate the initial potential energy stored in the compressed spring.
Step 2: Determine the total mechanical energy of the block-spring system at launch.
Step 3: Calculate the total energy lost due to denting, permanent deformation, and sound.
Step 4: Calculate the kinetic energy of the block at the end of the cannon.
Step 5: Determine the final temperature of the block.
Let's go through each step in detail:
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 stiffness of the spring and x is the compression distance.
In this case, the stiffness of the spring (k) is given as 250117 N/m, and the compression distance (x) is 1 meter.
Therefore, the initial potential energy (PE) stored in the spring is: PE = (1/2)(250117 N/m)(1 m)^2 = 125058.5 J.
Step 2: Determine the total mechanical energy of the block-spring system at launch.
The total mechanical energy (TE) of the system is the sum of initial potential energy (PE) and initial kinetic energy (KE), since the block is at rest initially.
Since the block is at rest, the initial kinetic energy is zero.
Therefore, the total mechanical energy (TE) at launch is: TE = PE + KE = 125058.5 J + 0 J = 125058.5 J.
Step 3: Calculate the total energy lost due to denting, permanent deformation, and sound.
According to the given information, 50% of the energy went into permanent deformation and 10% went to sound. This means that the remaining 40% of the energy is lost due to denting.
So, the total energy lost (EL) is: EL = 40% of TE = 0.4 * 125058.5 J = 50023.4 J.
Step 4: Calculate the kinetic energy of the block at the end of the cannon.
Assuming no further energy losses after leaving the cannon and neglecting air resistance, the total mechanical energy remains constant. Therefore, the kinetic energy (KEf) of the block at the end of the cannon is: KEf = TE - EL = 125058.5 J - 50023.4 J = 75035.1 J.
Step 5: Determine the final temperature of the block.
The final temperature of the block can be calculated using the principle of conservation of energy.
The energy lost due to heating (Q) can be expressed as: Q = m * (Cw) * ΔT, where m is the mass of the block (given as 10 grams = 0.01 kg), Cw is the specific heat capacity of water (4.18 J/g°C), and ΔT is the change in temperature.
Since the final temperature is given as 0 degrees Celsius, the change in temperature (ΔT) is: ΔT = Tf - Ti = 0°C - (-4°C) = 4°C.
Now, substituting the values into the equation: Q = 0.01 kg * (4.18 J/g°C) * 4°C = 0.1672 J.
The percent of the block melted is given by: (Q / KEf) * 100% = (0.1672 J / 75035.1 J) * 100% ≈ 0.0223%.
Therefore, the block has melted by approximately 0.0223%.