A metal sphere moving at a high speed v strikes a rigid target and comes to a complete halt. It is found that 80 % of its lost kinetic energy is absorbed by the sphere itself as heat. This absorbed heat is just enough to raise its temperature from 300 K to its melting point of 700 K and melt it completely. If the specific heat capacity of the metal is 250 J kg ⁻¹ K ⁻¹ and its latent heat of fusion is 4.4 10^4 J kg ⁻¹

Step-by-step solution

Let the mass of the sphere be m . The initial kinetic energy of the sphere is K = 1 2 m v^2 . The heat absorbed by the sphere is 80 % of its initial kinetic energy: Q = 0.8 ( 1 2 m v^2 ) = 0.4 m v^2 This heat is utilized in two stages: raising the temperature to the melting point (sensible heat) and melting the sphere completely (latent heat). Q = m c T + m L Equating the two expressions for Q : 0.4 m v^2 = m (c T + L) The mass m cancels out from both sides: 0.4 v^2 = c (T₂ - T₁) + L Substitute the given values ( c