How Fast Must a Bullet Travel to Melt on Impact?

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To determine the minimum nozzle velocity of a gun that causes a lead bullet to melt upon impact, the forensic investigator needs to calculate the total energy required to heat the bullet to its melting point and then melt it. The bullet, weighing 7.2g, requires 287.4J to reach the melting temperature of 327°C, and an additional 180J to melt, totaling 467.4J. This energy must be derived from the bullet's kinetic energy, which is given by the equation K = 1/2*m*v^2. By rearranging this equation to solve for velocity, the investigator can find the minimum nozzle velocity necessary for the bullet to melt on impact. Understanding the relationship between thermal energy and kinetic energy is crucial in solving this problem.
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Homework Statement


A forensic investigator notes a 7.2g lead bullet stopped in a door frame melted completely on impact. Assuming the bullet was shot at RT (20C) what would the investigator calculate as the minimum nozzle velocity of the gun to be? (C_lead = 130J/kg*C), (L_lead = 2.5*10^4J/kg) Lead melts at 327C


Homework Equations


Q = mc(delta)T
K = 1/2*m*v^2 (?)

The Attempt at a Solution


I know that the objective of this problem is to use the energy to determine the velocity. I found that Q = 287.4J. I'm not sure what to do from here, however. If I multiply 2.5*10^4J/kg by 0.0072kg, I get 180J. Do I subtract this from 287.4J, and use the result to calculate velocity? Is the difference kinetic energy? I'm not sure if I'm on the right track here. Any help is appreciated!
 
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What is the total energy needed to melt the bullet? Remember, it has to be first heated up to the melting point, and the total energy is equal to energy needed to heat it and then melt it. This energy has to come from the kinetic energy.
 

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