In a perfectly inelastic collision, two spheres made of the same material with masses 15 kg and 25 kg, moving in opposite directions with speeds of 10 m/s and 30 m/s, respectively, strike each other and stick together. The rise in temperature (in °C), if all the heat produced during the collision is retained by these spheres, is :
(specific heat of sphere material 31 cal/kg. °C and 1 cal = 4.2 J)

Question

In a perfectly inelastic collision, two spheres made of the same material with masses 15 kg and 25 kg, moving in opposite directions with speeds of 10 m/s and 30 m/s, respectively, strike each other and stick together. The rise in temperature (in °C), if all the heat produced during the collision is retained by these spheres, is :

(specific heat of sphere material 31 cal/kg. °C and 1 cal = 4.2 J)

G1 Physics · Accepted Answer

For a perfectly inelastic collision, e = 0. The kinetic energy lost is K(loss) = ½ μ v(rel)², where the reduced mass is μ = m₁m₂/(m₁ + m₂). Here, m₁ = 15 kg, m₂ = 25 kg, v(rel) = 10 + 30 = 40 m/s. Thus, μ = (15×25)/(15+25) = 375/40 = 9.375 kg. Therefore, K(loss) = ½(9.375)(40²) = 7500 J. The specific heat is S = 31 cal/(kg·°C) = 31×4.2 J/(kg·°C). The total mass is m(total) = 40 kg. If all lost kinetic energy becomes heat, 7500 = 40×31×4.2×ΔT. Hence, ΔT = 7500/(40×31×4.2) ≈ 1.44°C. Correct optio…