A 25,000-kg subway train initially traveling at 15.5 m / s slows to a stop in a station and then

Step 1: Calculate the initial kinetic energy of the subway train. u000AThe formula for kinetic ener

A 25,000-kg subway train initially traveling at 15.5 m / s...

A $25,000-\mathrm{kg}$ subway train initially traveling at 15.5 $\mathrm{m} / \mathrm{s}$ slows to a stop in a station and then stays there long enough for its brakes to cool. The station's dimensions are 65.0 $\mathrm{m}$ long by 20.0 $\mathrm{m}$ wide by 12.0 $\mathrm{m}$ high. Assuming all the work done by the brakes in stopping the train is transferred as heat uniformly to all the air in the station, by how much does the air temperature in the station rise? Take the density of the air to be 1.20 $\mathrm{kg} / \mathrm{m}^{3}$ and its specific heat to be 1020 $\mathrm{J} / \mathrm{kg} \cdot \mathrm{K}$ .

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Key Concepts

Density and Volume

Density, combined with the volume of a space, allows us to determine the total mass of the substance—in this case, the air in the station. This mass is then used alongside the specific heat capacity and the transferred energy to compute the temperature change using the formula Q = m * c * ?T.

Specific Heat Capacity

Specific heat capacity is a property of a substance that indicates how much energy is needed to raise the temperature of a unit mass of the substance by one degree Celsius (or Kelvin). It is used in the calculation that relates the absorbed heat energy to the resulting temperature increase of the air in the station.

Conservation of Energy

This concept states that energy is neither created nor destroyed, only transformed from one form to another. In the context of the problem, the kinetic energy of the moving train is fully converted into thermal energy when the train comes to a stop, demonstrating energy transfer from mechanical to heat energy.

Kinetic Energy

Kinetic energy is the energy possessed by an object due to its motion and is given by the formula (1/2)mv². In the problem, calculating the train’s kinetic energy is crucial because it quantifies the total amount of energy available for conversion into heat during braking.

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