A 19-g lead bullet is shot with a speed of 340 m/s into a wooden wall. Assuming that 70% of the kinetic energy is absorbed by the bullet as heat (and 30% by the wall), what is the final temperature of the bullet? (Assume the bullet is initially at room temperature of 20°C. Use the numerical data found in this table.)
I know the answer is 334 degree C. How do I get this answer?
I tried using:
.7E = 1/2mv^2 - cm(dT)
However, I do not know the total energy of the system, so I can't finish the problem this way.
The table gives the specific heat of lead as 0.129 KJ/(kgK) and 0.0308 cal/(gK).
0.7•KE=Q
0.7•mv²/2=mc(tº-20º)
tº= 20º+0.7•v²/2 =
=20º+0.7•340²/2•129= 333.6ºC≈334ºC
To determine the final temperature of the bullet, we need to consider the conservation of energy. The kinetic energy of the bullet is converted into heat energy, which causes the temperature of the bullet to rise.
Here's the step-by-step approach to calculate the final temperature:
Step 1: Find the initial kinetic energy of the bullet.
The kinetic energy (KE) of an object can be calculated using the formula:
KE = 0.5 * mass * velocity^2
Given:
Mass of the bullet (m) = 19 g = 0.019 kg
Initial velocity (v) = 340 m/s
Using the formula, substitute the values to find the initial kinetic energy of the bullet:
KE_initial = 0.5 * 0.019 kg * (340 m/s)^2
Step 2: Calculate the energy absorbed by the bullet.
As stated in the problem, 70% of the kinetic energy is absorbed by the bullet as heat. Therefore, multiply the initial kinetic energy by 0.7 to find the energy absorbed by the bullet:
Energy_absorbed = KE_initial * 0.7
Step 3: Calculate the energy absorbed by the wall.
Similarly, 30% of the kinetic energy is absorbed by the wall. Multiply the initial kinetic energy by 0.3 to find the energy absorbed by the wall:
Energy_absorbed_wall = KE_initial * 0.3
Step 4: Calculate the heat required to raise the temperature of the bullet.
To find the heat required to raise the temperature of the bullet, we can use the formula:
Q = m * c * ΔT
Given:
m = mass of the bullet = 0.019 kg
c = specific heat capacity of lead = 130 J/kg°C (from the given table)
ΔT = change in temperature
Rearranging the equation, we can solve for ΔT:
ΔT = Q / (m * c)
Substitute the values:
ΔT = Energy_absorbed / (m * c)
Step 5: Calculate the final temperature.
The final temperature can be found by adding the change in temperature (ΔT) to the initial temperature.
Given:
Initial temperature (T_initial) = 20°C
Final Temperature (T_final) = T_initial + ΔT
Now, substitute the values and calculate T_final:
T_final = 20°C + ΔT
By following these steps and performing the calculations, you should arrive at the final temperature of the bullet, which is 334°C.