Two identical lead balls are moving towards each other at the same speed. At what speed can they melt

Two identical lead balls are moving towards each other at the same speed. At what speed can they melt in a head-on collision? The initial temperature of the balls is 27 C, do not take into account the heat transfer between the balls and the environment.

Given:

t0 = 27 ° Celsius – the initial temperature of the lead balls;

t = 327 ° Celsius – melting point of lead;

q = 25000 Joule / kilogram – specific heat of fusion of lead;

c = 130 J / (kg * C) – specific heat capacity of lead.

It is required to determine v (m / s) – with what speed two identical lead balls must move in order for them to completely melt upon impact.

Let m be the mass of the lead balls. Then, the total kinetic energy of these balls should be enough to heat and melt these balls, that is:

Qkinetic = Qheating + Qmelting;

m * v ^ 2/2 = c * m * (t – t0) + q * m;

v ^ 2/2 = c * (t – t0) + q;

v ^ 2 = 2 * c * (t – t0) + 2 * q;

v = (2 * c * (t – t0) + 2 * q) ^ 0.5 = (2 * 130 * 300 + 2 * 25000) ^ 0.5 = (78000 + 50,000) ^ 0.5 = = 128000 ^ 0.5 = 357 m / s.

Answer: the balls should move at a speed of 357 m / s (the result is slightly different from the one required in the condition, since the specific heat of melting of lead is present in the solution of the problem, which depends on its type and differs in different reference books)