CP What must the initial speed of a lead bullet be at 25.0o C so that the heat developed when it is brought to rest will be just sufficient to melt it? Assume that all the initial mechanical energy of the bullet is converted to heat and that no heat flows from the bullet to its surroundings. (Typical rifles have muzzle speeds that exceed the speed of sound in air, which is 347 m/s at 25.0o C.)
Solution 51E Let the initial speed of the lead bullet be x. Therefore, its kinetic energy is = mx …..(1), where m is the mass of the bullet. 2 0 Melting point of lead is = 327.5 C Initial temperature = 25 C 0 0 0 0 Therefore, the change in temperature is T = 327.5 25 C = 302.5 C Specific heat of lead s = 128 J/kg C 0 0 0 Heat generated during melting of the bullet Q = m 1 128 J/kg C × 302.5 C Q 1 38720m J Latent heat of fusion of lead L = 23000 J/kg Heat generated when the bullet changes its phase Q = m × 23000 J Now, the kinetic energy of the bullet will be equal to the summation of Q and Q . 1 2 1 2 Therefore, mx2= 38720m + 23000m x = 2 × (38720 + 23000) 2 x = 123440 x = 351 m/s Therefore, the initial speed of the bullet will be approximately 351 m/s.