A 8.2 gram lead bullet that was stopped in a doorframe apparently melted completely on impact. Assuming the bullet was fired at room temp., what is the minimum muzzle velocity of the gun?
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To find the minimum muzzle velocity of the gun, we need to use principles of conservation of energy and assumptions about the bullet's heat capacity and melting point. Here's the step-by-step approach to finding the answer:
1. Determine the energy required to melt the bullet:
The energy needed to melt a substance can be calculated using the equation Q = m * ΔH, where Q is the energy, m is the mass, and ΔH is the heat of fusion (energy per unit mass required to change a substance from solid to liquid state). For lead, the heat of fusion is approximately 24.5 J/g.
Q = (mass of bullet) * (heat of fusion)
Q = 8.2 g * 24.5 J/g
Q ≈ 200.9 J
2. Calculate the kinetic energy of the bullet:
The kinetic energy of an object is given by the equation KE = (1/2) * m * v^2, where KE is the kinetic energy, m is the mass, and v is the velocity.
KE = (1/2) * (mass of bullet) * (velocity of bullet)^2
Since we want to find the minimum velocity (in order to have enough kinetic energy to melt the bullet), we'll use the minimum mass for the bullet (8.2 g).
KE = (1/2) * 8.2 g * (velocity of bullet)^2
3. Equate the energy required to melt the bullet to the kinetic energy of the bullet and solve for the velocity:
Set the energy required to melt the bullet equal to the kinetic energy of the bullet and solve for the velocity (v).
(1/2) * 8.2 g * (velocity of bullet)^2 = 200.9 J
Rearranging the equation:
(velocity of bullet)^2 = (2 * 200.9 J) / 8.2 g
velocity of bullet ≈ √(492) ≈ 22.2 m/s
Therefore, the minimum muzzle velocity of the gun is approximately 22.2 m/s.