It has been said that sometimes lead bullets melt upon impact. Assume that a bullet with an initial temperature of 295 K receives 73.5 % of the work done on it by a wall on impact as an increase in internal energy. (The melting point of lead is 601 K, the heat of fusion for lead is L = 23.2 kJ/kg, and the specific heat of lead is c = 0.129 kJ/kg K.)
a) What is the minimum speed with which a 12.9-g lead bullet would have to hit a surface (assuming the bullet stops completely and all the kinetic energy is absorbed by it) in order to begin melting?
b) What is the minimum impact speed required for the bullet to melt completely?
1/2 m v^2 * .735=mcDtemp+mHf
divide out mass m
1/2 v^2*.735=.129(601-295)+23.2
solve for v all melting
for betinning melting, just the first trem of the right.
To begin solving this problem, we need to analyze the energy transfer that occurs during the impact of the bullet with the surface.
a) The minimum speed required for the bullet to begin melting can be calculated by considering the increase in internal energy due to the work done on the bullet and the energy required for the phase change from solid to liquid.
1. First, let's calculate the work done on the bullet by the wall. We know that 73.5% of this work is absorbed as an increase in internal energy. Therefore, we have:
Work done on bullet = Change in internal energy = 73.5% of total work done
2. The work done on the bullet can be calculated using the formula:
Work = Force × Distance
Since the bullet stops completely, the distance it travels is equal to the displacement of the bullet.
3. Now let's calculate the change in internal energy of the bullet using the formula:
Change in internal energy = m × c × ΔT
where m is the mass of the bullet, c is the specific heat of lead, and ΔT is the change in temperature.
4. The change in temperature can be calculated as:
ΔT = final temperature - initial temperature
Since the bullet will melt at its melting point, the final temperature will be 601 K.
5. Finally, we can calculate the minimum speed required by equating the work done on the bullet to the change in internal energy:
Work done on bullet = Change in internal energy
Let's proceed with the calculations:
Given:
Initial temperature (Ti) = 295 K
Final temperature (Tf) = 601 K
Melting point of lead = 601 K
Specific heat of lead (c) = 0.129 kJ/kg K
Mass of the bullet (m) = 12.9 g = 0.0129 kg
Heat of fusion for lead (L) = 23.2 kJ/kg
1. Calculating the work done on the bullet:
Work done = Work = Force × Distance
Since the bullet stops completely, we have:
Distance = Displacement of the bullet
2. Calculating the change in internal energy:
Change in internal energy = m × c × ΔT
ΔT = Tf - Ti
3. Equating the work done on the bullet to the change in internal energy:
Work done = Change in internal energy
Now, let's plug in the values and perform the calculations:
Work done = Force × Distance
Change in internal energy = m × c × ΔT
Work done = Change in internal energy
Let's rearrange the equation and solve for the minimum speed:
Force × Distance = m × c × ΔT
Force = m × c × ΔT / Distance
Since we are considering the minimum speed, the force should be the maximum force that the wall can exert on the bullet.
Force = Maximum force from the wall = mass of the bullet × acceleration due to gravity (g)
Substituting this back into the equation:
m × c × ΔT / Distance = m × g
To find the minimum speed, we can use the equation for kinetic energy:
KE = 1/2 × m × v^2
Where KE is the kinetic energy, m is the mass of the bullet, and v is the velocity of the bullet.
Assuming all the kinetic energy is absorbed by the bullet, the kinetic energy should be equal to the work done on the bullet.
KE = Work done
Therefore:
1/2 × m × v^2 = m × c × ΔT / Distance
Simplifying the equation:
v^2 = 2 × c × ΔT / Distance
Now we can solve for the minimum speed:
v = sqrt(2 × c × ΔT / Distance)
Let's plug in the given values:
ΔT = Tf - Ti = 601 K - 295 K
Distance = Displacement of the bullet
After substituting these values and performing the calculations, we can find the minimum speed required for the bullet to begin melting.
To answer these questions, we can use the principles of work and energy, along with the concepts of heat transfer and the first law of thermodynamics.
Let's start by understanding the given information and the steps involved in finding the answers to both parts, a) and b):
Given:
- Initial temperature of the lead bullet (T_initial) = 295 K
- Work done on the bullet by the wall as an increase in internal energy = 73.5% (0.735)
- Melting point of lead (T_melting) = 601 K
- Heat of fusion for lead (L) = 23.2 kJ/kg
- Specific heat of lead (c) = 0.129 kJ/kg K
Steps:
a) To find the minimum speed required for the bullet to begin melting:
1) Calculate the change in internal energy (ΔU) of the bullet using the work done on it.
2) Determine the amount of heat required to raise the temperature of the bullet from its initial temperature to the melting point.
3) Equate the change in internal energy with the heat required for this temperature change.
4) Calculate the kinetic energy of the bullet using its mass and velocity. Equate this kinetic energy to the heat required.
5) Solve the equation to find the minimum speed required for the bullet to begin melting.
b) To find the minimum impact speed required for the bullet to completely melt:
1) Use the same method as in part a) to determine the minimum speed required for the bullet to begin melting.
2) Calculate the additional heat required to completely melt the bullet.
3) Set up an equation equating the total heat required to the initial kinetic energy of the bullet.
4) Solve the equation to find the minimum impact speed required for complete melting.
Let's now go through the calculations step by step for both parts:
a) Minimum speed to begin melting:
1) Change in internal energy (ΔU):
ΔU = Work done on the bullet
ΔU = 0.735 * Work done on the bullet
2) Heat required for temperature change:
q = mcΔT
q = mass * specific heat * (T_final - T_initial)
3) Equating the change in internal energy with the heat required:
ΔU = q
0.735 * Work done on the bullet = mass * specific heat * (T_final - T_initial)
4) Equating kinetic energy to heat required:
0.5 * mass * v^2 = mass * specific heat * (T_final - T_initial)
5) Solving for the minimum speed (v):
v = sqrt((2 * specific heat * (T_final - T_initial)) / 0.5)
b) Minimum impact speed for complete melting:
Follow the same steps as part a), but include the additional heat required for complete melting.
1) Calculate the minimum speed to begin melting.
2) Calculate the additional heat required to completely melt the bullet:
additional heat = mass * heat of fusion
3) Set up an equation equating the total heat required to the initial kinetic energy of the bullet:
(total heat required) = (initial kinetic energy)
4) Solve the equation to find the minimum impact speed required for complete melting.
With these steps and calculations, you should be able to find the answers to both parts, a) and b), of the question.