Final answer:
To calculate the minimum muzzle velocity, we would equate the heat energy needed to raise the bullet's temperature to its melting point and melt it with the bullet's kinetic energy at impact, then solve for the velocity.
Step-by-step explanation:
The question requires an application of physics principles, specifically the concepts of energy transfer and change of state, to calculate the minimum muzzle velocity of a gun based on the melting of a bullet upon impact. The process involves several steps:
- Calculate the amount of heat energy needed to raise the bullet's temperature to its melting point and then melt it.
- Assume that all kinetic energy of the bullet at the moment of impact is converted into heat energy.
- Use the kinetic energy formula to solve for the muzzle velocity.
We start by determining the heat energy required to heat the lead bullet from room temperature (20°C) to its melting point and then melt it. The specific heat capacity of lead (c) and its heat of fusion (Λ) are needed for this calculation. The specific heat capacity of lead is approximately 0.128 J/g°C, and the heat of fusion is approximately 24.7 J/g. With this information, we can calculate the total heat energy (Q) required using the formula:
Q = mcΔT + mΛ
where m is the mass of the bullet, c is the specific heat capacity, ΔT is the change in temperature, and Λ is the heat of fusion.
Next, we equate the total heat energy Q to the kinetic energy (KE) of the bullet just before impact, which is given by:
KE = ½mv²
where m is the mass of the bullet and v is the velocity.
Finally, we rearrange the kinetic energy equation to solve for the bullet's muzzle velocity (v) and find the minimum velocity required for the bullet to have enough kinetic energy to melt upon impact.