Ch. 19 - Heat and the First Law of Thermodynamics

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook

Giancoli Douglas 5th edition

Ch. 19 - Heat and the First Law of Thermodynamics

Problem 29

Chapter 19, Problem 29

At a crime scene, the forensic investigator notes that the 6.2-g lead bullet that was stopped in a doorframe apparently melted completely on impact. Assuming the bullet was shot at room temperature (20°C), what does the investigator calculate as the minimum muzzle velocity of the gun?

Verified step by step guidance

Determine the total energy required to melt the bullet. This includes the energy needed to raise the temperature of the lead bullet from room temperature (20°C) to its melting point, and the energy required to melt it completely. Use the formula for heat transfer:

Q = mcΔT

for the temperature increase, and

Q = mL

for the phase change, where

m

is the mass,

c

is the specific heat capacity,

ΔT

is the temperature change, and

L

is the latent heat of fusion.

Look up the necessary physical constants for lead: specific heat capacity (

c = 128 \, \(\text{J/kg·°C}\)

), melting point (

T_m = 327.5 \, \(\text{°C}\)

), and latent heat of fusion (

L = 2.45 \(\times\) 10^4 \, \(\text{J/kg}\)

Calculate the energy required to raise the temperature of the bullet to its melting point using

Q_1 = mcΔT

, where

ΔT = T_m - T_i

(initial temperature is 20°C).

Calculate the energy required to melt the bullet completely using

Q_2 = mL

. Add

Q_1

and

Q_2

to find the total energy

Q_{\(\text{total}\)}

required to melt the bullet.

Relate the total energy to the kinetic energy of the bullet using the formula for kinetic energy:

KE = \(\frac{1}{2}\)mv^2

. Solve for the minimum muzzle velocity

v

by equating

KE = Q_{\(\text{total}\)}

and rearranging to find

v = \(\sqrt{\frac{2Q_{\text{total}\)}}{m}}

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Kinetic Energy

Kinetic energy is the energy possessed by an object due to its motion, calculated using the formula KE = 1/2 mv², where m is mass and v is velocity. In this scenario, the kinetic energy of the bullet at the moment of impact is crucial for determining how much energy was available to cause the bullet to melt upon hitting the doorframe.

Heat Transfer and Melting

Heat transfer refers to the movement of thermal energy from one object to another, which can occur through conduction, convection, or radiation. When the bullet strikes the doorframe, its kinetic energy is converted into thermal energy, raising the bullet's temperature to its melting point, which is essential for calculating the energy required for the bullet to melt.

Conservation of Energy

The principle of conservation of energy states that energy cannot be created or destroyed, only transformed from one form to another. In this context, the kinetic energy of the bullet is transformed into thermal energy upon impact, and understanding this principle allows the investigator to relate the bullet's initial velocity to the energy needed for melting.

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