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Question #42522

Difficulty: 8/10 | 🟢 Free

The Question

A projectile of mass $m$ moving with velocity $v_0$ strikes a block of ice of mass $M$ (initially at rest at $0^\circ\text{C}$ ) and gets embedded in it. The collision takes place in an environment at constant atmospheric pressure $P_0$ . The specific internal energy of fusion of ice (change in internal energy per unit mass during melting) is $u_f$ . The densities of ice and water are $\rho_i$ and $\rho_w$ respectively ( $\rho_i < \rho_w$ ). Assuming no heat is exchanged with the surroundings and all the dissipated mechanical energy is retained within the ice-projectile system, what is the mass of ice $\Delta m$ that melts?

Options

A.
$\frac{m M v_0^2}{2(m+M) \left[ u_f - P_0 \left( \frac{1}{\rho_i} - \frac{1}{\rho_w} \right) \right]}$
B.
$\frac{m M v_0^2}{2(m+M) u_f}$
C.
$\frac{m M v_0^2}{2(m+M) \left[ u_f + P_0 \left( \frac{1}{\rho_i} - \frac{1}{\rho_w} \right) \right]}$
D.
$\frac{m v_0^2}{2 \left[ u_f - P_0 \left( \frac{1}{\rho_i} - \frac{1}{\rho_w} \right) \right]}$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Latent Heat and Volume Change|Collision Impulses|Vector Cross-Products

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of collision mechanics and the First Law of Thermodynamics.

Students commonly assume that the lost kinetic energy during an inelastic collision is entirely and exclusively equated to the internal energy change of the phase transition.

The trap lies in neglecting the $P \Delta V$ work done by the atmosphere.

Because ice contracts upon melting ( $\rho_i < \rho_w$ ), the atmosphere does positive work on the system, which contributes to the internal energy required for fusion.

Failing to account for this work leads to an incorrect energy balance.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Thermal Properties of Matter - Class 11 Part 2
Concept	Latent Heat and Volume Change
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free