At a crime scene, the forensic investigator notes that the 7.0-g lead bullet that was stopped in a doorframe apparently melted completely on impact. Assuming the bullet was fired at room temperature (20°C), what does the investigator calculate as the minimum muzzle velocity of the gun?

To calculate the minimum muzzle velocity of the gun, we need to apply the principle of conservation of energy. Since the bullet melted completely on impact, we can assume that all of its kinetic energy (KE) was converted into heat energy (Q) to melt the bullet.

First, we need to determine the amount of heat energy required to melt the bullet. We can use the specific heat capacity (c) of lead, which is approximately 0.13 J/g°C, and the melting point (Tm) of lead, which is 327°C. The formula to calculate the heat energy required to melt the bullet is:

Q = m * c * ΔT

Where:
Q = Heat energy required (in Joules)
m = Mass of the bullet (in grams)
c = Specific heat capacity of lead (in J/g°C)
ΔT = Change in temperature = Tm - room temperature

Calculating the change in temperature:
ΔT = Tm - room temperature
ΔT = 327°C - 20°C
ΔT = 307°C

Now, we can calculate the heat energy required:
Q = 7.0g * 0.13 J/g°C * 307°C
Q ≈ 282.91 Joules

The heat energy is equal to the kinetic energy (KE) of the bullet before impact. The formula to calculate kinetic energy is:

KE = (1/2) * m * v^2

Where:
KE = Kinetic energy (in Joules)
m = Mass of the bullet (in kilograms)
v = Velocity of the bullet (in meters per second)

To convert the mass of the bullet from grams to kilograms, we divide by 1000:
m = 7.0g / 1000
m = 0.007 kg

Now, rearrange the equation to solve for velocity:
v = √((2 * KE) / m)

Plugging in the values:
v = √((2 * 282.91) / 0.007)
v ≈ 596.34 m/s

Therefore, the investigator would calculate the minimum muzzle velocity of the gun to be approximately 596.34 m/s.