Explanation

Let the mass $m_1 = 15\,\text{kg}$ move with velocity $v_1 = +10\,\text{m/s}$ and mass $m_2 = 25\,\text{kg}$ move with velocity $v_2 = -30\,\text{m/s}$.

Since the collision is perfectly inelastic, the two spheres stick together after collision.

Using conservation of linear momentum,

$$ m_1 v_1 + m_2 v_2 = (m_1+m_2)v $$

$$ 15(10) + 25(-30) = 40v $$

$$ 150 - 750 = 40v $$

$$ v = -15\,\text{m/s} $$

Initial kinetic energy of the system is

$$ KE_i = \frac12 m_1 v_1^2 + \frac12 m_2 v_2^2 $$

$$ KE_i = \frac12(15)(10^2) + \frac12(25)(30^2) $$

$$ KE_i = 750 + 11250 = 12000\,\text{J} $$

Final kinetic energy after collision is

$$ KE_f = \frac12 (m_1+m_2) v^2 $$

$$ KE_f = \frac12 (40)(15^2) = 4500\,\text{J} $$

Loss in kinetic energy is converted into heat,

$$ Q = KE_i - KE_f = 12000 - 4500 = 7500\,\text{J} $$

Convert heat into calories,

$$ Q = \frac{7500}{4.2} \approx 1786\,\text{cal} $$

Total mass of spheres is $40\,\text{kg}$ and specific heat is $31\,\text{cal/kg·}^\circ\text{C}$.

Using heat relation,

$$ Q = mc\Delta T $$

$$ 1786 = 40 \times 31 \times \Delta T $$

$$ \Delta T \approx 1.44^\circ\text{C} $$

Hence, the rise in temperature is

$$ \boxed{1.44^\circ\text{C}} $$

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