Question #42158

A thermally insulated vessel contains a mixture of water and $1 \text{ kg}$ of ice at $0^\circ\text{C}$. The total mass of the vessel and its contents is $8 \text{ kg}$. A projectile of mass $2 \text{ kg}$ traveling horizontally at $1000 \text{ m/s}$ strikes the vessel and embeds within it. The vessel is free to move on a frictionless horizontal surface. The collision is perfectly inelastic and generates heat internally within the vessel's thermal control volume. Assume all lost mechanical energy is instantaneously converted to heat. The total heat capacity of the system (vessel, mixture, and embedded projectile) is $C = 10,000 \text{ J/}^\circ\text{C}$ (assumed constant, excluding the latent heat of ice). The latent heat of fusion of ice is $L_f = 3.2 \times 10^5 \text{ J/kg}$. Immediately after the ice melts and thermal equilibrium is reached, the vessel's insulation is removed, exposing it to surroundings at $0^\circ\text{C}$. The system then cools according to Newton's Law of Cooling with a cooling constant $k = 0.1 \text{ min}^{-1}$. What is the temperature of the system $10 \ln 2$ minutes after the insulation is removed?