When a lead bullet hits a solid target at high speed, its entire kinetic energy is converted into heat. For simplicity, assume that all the heat is concentrated in the bullet rather than the target.
If the bullet's initial temp. is 20` C, how fast should it be moving before the collision in order to reach the lead's melting temp 327.3` C and have 53% of its mass melted after the collision? Lead's specific heat is 128 J/kg x K and it's latent heat of fusion is 24500 J/kg.
answers is in m/s?????
Set the bullet's kinetic energy
(1/2) MV^2
equal to the amount of heat energy required to do that amount of heating and melting. You do not need to know the mass M; it will cancel out later.
Both the kinetic energy and the energy needed to melt 53% of the lead are proportional to M
i'm still confused.... :(
Try to follow the steps.
(1) compute the energy (per mass) required to raise the lead tempertaure and melt 53% of it.
(2) Set that energy equal to the kinetic energy (per mass), and solve the resulting equation for the velocity required.
To solve this problem, we need to consider the energy transformations involved and calculate the bullet's velocity needed to reach the desired melting temperature and have 53% of its mass melted after the collision.
1. Calculate the thermal energy required to melt 53% of the bullet's mass:
- The bullet's mass that needs to be melted = 53% of the bullet's total mass
- The thermal energy required = mass x specific heat x change in temperature
- Change in temperature = melting temperature - initial temperature
2. Calculate the energy required to raise the bullet's temperature to the melting temperature:
- The thermal energy required = mass x specific heat x change in temperature
3. Calculate the kinetic energy of the bullet:
- The kinetic energy = (1/2) x mass x velocity^2
4. Equate the thermal energy required to melt the bullet and the kinetic energy:
- Thermal energy required to melt = kinetic energy
5. Rearrange the equation and solve for the velocity (v):
- v = √[(2 x thermal energy required) / mass]
Let's apply these steps and solve the problem:
Given data:
Initial temperature (T1) = 20` C
Melting temperature (T2) = 327.3` C
Mass of the bullet (m) = Total mass
Specific heat (c) = 128 J/kg x K
Latent heat of fusion (L) = 24500 J/kg
Step 1:
Mass that needs to be melted (m_melted) = 53% x m
Thermal energy required to melt = m_melted x L
Change in temperature = T2 - T1
Step 2:
Thermal energy required to raise the temperature to T2 = m x c x change in temperature
Step 3:
Kinetic energy (KE) = (1/2) x m x v^2
Step 4:
Thermal energy required to melt = KE
Step 5:
v = √[(2 x thermal energy required) / m]
Now, plug in the given values and calculate the required velocity (v) in meters per second:
- Convert the percentage to decimal:
m_melted = 0.53 x m
- Calculate thermal energy required to melt:
Thermal energy required to melt = m_melted x L
- Calculate change in temperature:
change in temperature = T2 - T1
- Calculate thermal energy required to raise the temperature:
Thermal energy required to raise the temperature = m x c x change in temperature
- Equate thermal energy required to melt and kinetic energy:
2 x thermal energy required = KE
- Rearrange the equation to solve for velocity:
v = √[(2 x (Thermal energy required to melt + Thermal energy required to raise the temperature)) / m]
Substitute the calculated values and solve for velocity (v).
Note: Since the specific mass (m) and total mass are not provided, you will need to input those values to calculate the actual velocity in m/s.