A gun is a heat engine. In particular, it is an internal...

A gun is a heat engine. In particular, it is an internal combustion piston engine that does not operate in a cycle, but comes apart during its adiabatic expansion process. A certain gun consists of $1.80 \mathrm{kg}$ of iron. It fires one 2.40 -g bullet at $320 \mathrm{m} / \mathrm{s}$ with an energy efficiency of $1.10 \% .$ Assume the body of the gun absorbs all the energy exhaust-the other $98.9 \%$ -and increases uniformly in temperature for a short time interval before it loses any energy by heat into the environment. Find its temperature increase.

Key Concepts

Specific Heat Capacity

Specific heat capacity is a material property that indicates the amount of energy required to raise the temperature of a unit mass by one degree Celsius. It is vital for calculating how absorbed thermal energy affects temperature changes in objects, such as determining the increase in the gun's temperature from the waste heat.

Energy Conversion and Efficiency

This concept involves the transformation of energy from one form to another, such as converting the chemical energy stored in gunpowder into the kinetic energy of a bullet. Efficiency quantifies the proportion of energy that is usefully transformed, with the remaining energy typically wastefully converted into heat. Understanding efficiency is key to analyzing systems where energy is partitioned into useful work and heat loss.

Conservation of Energy

The conservation of energy principle states that the total energy in an isolated system remains constant, though it may change forms. In the context of a gun acting as a heat engine, the total chemical energy is split between the kinetic energy of the bullet and the thermal energy absorbed by the gun's body. This balance is crucial in evaluating the energy available for raising the temperature of the gun.

Kinetic Energy

Kinetic energy is the energy possessed by an object due to its motion and is typically calculated using the formula ½mv². In scenarios like a bullet being fired, determining the kinetic energy is essential for understanding how much of the chemical energy is converted into mechanical work versus how much is left over to raise the temperature of other components.

Adiabatic Process

An adiabatic process is one in which no heat is exchanged with the surroundings. Although actual systems may eventually lose heat, during the rapid expansion in a gun the process can be assumed to be adiabatic. This assumption allows us to directly relate the absorbed energy to the temperature change in the gun’s material without considering external heat loss.

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