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When objects move, they possess energy called kinetic energy (KE). In the case of the ice-skater, this energy is transferred into heat energy due to friction when they come to a stop. To compute this heat energy, we first calculate the skater's kinetic energy using the formula \( KE = \frac{1}{2}mv^2 \), where \( m \) is the mass of the skater and \( v \) is their velocity. Friction converts all of this kinetic energy into heat (\( Q_{\text{total}} = KE \)).
However, not all the generated heat contributes to melting the ice; only a portion of it is absorbed by the ice. In the exercise, it's specified that 50\(\%\) of the generated heat is absorbed by the ice, defining the equation \( Q_{\text{ice}} = 0.5 \times KE \). This absorbed heat then plays its part in the subsequent melting of the ice.

The latent heat of fusion is a key concept when discussing phase changes. It's the amount of heat energy required to change a substance from solid to liquid without a change in temperature. For ice, this value is given as \( 334,000 \text{ J/kg} \).
In this exercise, once the heat energy is absorbed by the ice due to friction, it doesn't immediately heat up the ice. Instead, it uses that energy to melt, signifying a phase change from solid to liquid. This happens at a constant temperature, as indicated by the concept of latent heat. Understanding this concept is crucial in predicting how much ice will melt based on the absorbed energy. The relationship is described by the equation \( Q_{\text{ice}} = m \times L \), where \( m \) is the mass of the ice melted.

The final part of the exercise involves calculating how much ice actually melts. Using the absorbed heat \( Q_{\text{ice}} \) calculated from earlier, we apply the formula for latent heat: \( Q_{\text{ice}} = m \times L \). Here, \( L \) is the latent heat of fusion, a constant \( 334,000 \text{ J/kg} \), and \( m \) is the mass of ice melted.
To find the mass \( m \), we rearrange the equation to \( m = \frac{Q_{\text{ice}}}{L} \). This formula helps determine the quantity of ice that transforms from solid to liquid using the portion of heat it absorbs. Breaking down problems like this step-by-step instills a clear understanding of physical processes and how energy transfers function during them.