It has been said that sometimes lead bullets melt upon impact. Assume that a bullet with an initial temperature of 301 K receives 73.9 % 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 14.1-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?
To solve part a) and b), we need to consider the conservation of energy. The initial kinetic energy of the bullet is converted into work done against the bullet's path (W = Fd) and an increase in internal energy (ΔU = Q - W), where Q is the heat transferred to the bullet.
Let's break down the problem step by step:
Step 1: Calculate the initial kinetic energy (KE_initial) of the bullet.
The formula for kinetic energy is KE = (1/2)mv^2, where m is the mass of the bullet and v is its velocity.
Given:
Mass of the bullet (m) = 14.1 g = 0.0141 kg.
We need to find the minimum velocity, so we'll set the initial kinetic energy equal to zero:
KE_initial = 0
Step 2: Calculate the work done on the bullet by the wall.
According to the problem statement, 73.9% of the work on impact is converted into an increase in internal energy. This means:
W = (100% - 73.9%) * Work done on the bullet
Step 3: Calculate the increase in internal energy of the bullet.
ΔU = Q - W
Step 4: Calculate the final temperature of the bullet.
The final temperature of the bullet is the melting point of lead (601 K) since it begins to melt.
Now, let's solve part a) and b) using these steps:
a) To find the minimum speed required to begin melting, we need to find the velocity (v) when the bullet starts to melt. In this case, the increase in internal energy (ΔU) equals the heat of fusion (L) for lead.
ΔU = L = mcΔT, where ΔT is the change in temperature.
Since the bullet initially has a temperature of 301 K and reaches the melting point of lead (601 K), the change in temperature is ΔT = 601 K - 301 K = 300 K.
Using the specific heat formula, we have:
c = ΔU / (m * ΔT)
Solving for ΔU, we get:
ΔU = c * m * ΔT
Substituting the values:
L = c * m * ΔT
Rearranging the equation for v, we get:
v = sqrt(2L / m)
Substituting the given values:
v = sqrt(2 * (23.2 kJ/kg) / (0.0141 kg))
Calculating the result gives us the minimum velocity needed for the bullet to start melting.
b) To find the minimum impact speed required for the bullet to melt completely, we need to find the velocity (v) when the increase in internal energy (ΔU) is equal to the total work done on the bullet (W).
Using the equation ΔU = Q - W, we can set ΔU = W and solve for v.
Substituting the given values:
W = (100% - 73.9%) * Work done on the bullet
Rearranging the equation for v, we get:
v = sqrt((2mL) / (m(1 - 0.739)))
Simplifying further:
v = sqrt((2L) / (1 - 0.739))
Calculating the result gives us the minimum impact speed required for the bullet to completely melt.
Note: Make sure to convert the units appropriately to maintain consistency.
To solve this problem, we need to consider the energy balance and heat transfer involved in the bullet's impact and melting process.
a) To find the minimum speed required for the bullet to begin melting, we can equate the work done on the bullet to the increase in its internal energy:
Work done on the bullet = Increase in internal energy
The work done on the bullet is equal to the kinetic energy of the bullet. The increase in internal energy can be calculated using the formula:
ΔU = mcΔT + mL
Where:
ΔU = Increase in internal energy
m = Mass of the bullet
c = Specific heat of lead
ΔT = Change in temperature
L = Heat of fusion for lead
Given values:
m = 14.1 g = 0.0141 kg
c = 0.129 kJ/kg K
ΔT = (melting point of lead) - (initial temperature of the bullet) = 601 K - 301 K = 300 K
L = 23.2 kJ/kg = 23.2 * 10^3 J/kg
Substituting the values into the equation:
73.9% * (Kinetic energy of the bullet) = (0.0141 kg) * (0.129 kJ/kg K) * (300 K) + (0.0141 kg) * (23.2 kJ/kg)
Let's solve for the kinetic energy of the bullet:
Kinetic energy of the bullet = (73.9% * (0.0141 kg) * (0.129 kJ/kg K) * (300 K) + (0.0141 kg) * (23.2 kJ/kg)) / 73.9%
b) To find the minimum impact speed required to completely melt the bullet, we can equate the kinetic energy of the bullet to the sum of the increase in internal energy and the heat of fusion:
Kinetic energy of the bullet = Increase in internal energy + Heat of fusion
Using the same values as above, we can solve for the minimum impact speed:
(Kinetic energy of the bullet) = (0.0141 kg) * (0.129 kJ/kg K) * (300 K) + (0.0141 kg) * (23.2 kJ/kg)
Now, you can calculate the minimum speed and minimum impact speed by substituting the given values into the equations and performing the calculations.