/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 102 A 20.0 -g lead bullet leaves a r... [FREE SOLUTION] | 91影视

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Chapter 14: Problem 102

A 20.0 -g lead bullet leaves a rifle at a temperature of $47.0^{\circ} \mathrm{C}\( and travels at a velocity of \)5.00 \times 10^{2} \mathrm{m} / \mathrm{s}\( until it hits a large block of ice at \)0^{\circ} \mathrm{C}$ and comes to rest within it. How much ice will melt?

Short Answer

Expert verified
Answer: To find the amount of ice that melts, follow these steps: 1. Calculate the initial heat energy of the bullet: \(Q_{bullet} = (0.020 \text{kg})\times(128 \frac{\text{J}}{\text{kg} \cdot \text{K}}) \times (47 \text{K})\) 2. Calculate the initial kinetic energy of the bullet: \(KE_{bullet} = 0.5\times(0.020 \text{kg})\times(5.00\times10^2\frac{\text{m}}{\text{s}})^2\) 3. Calculate the total energy transferred to the ice: \(E_{total} = Q_{bullet} + KE_{bullet}\) 4. Determine the amount of ice that melts: \(m_{ice} = \frac{E_{total}}{L_f}\), where \(L_f = 3.33\times10^5 \frac{\text{J}}{\text{kg}}\) By substituting the values calculated in steps 1-3 into the formula in step 4, you will find the mass of ice that melts.

Step by step solution

01

Calculate the initial heat energy of the bullet

First, we need to calculate the heat energy of the bullet due to its initial temperature. The heat energy can be calculated using the formula \(Q = mc\Delta T\), where \(Q\) is the heat energy, \(m\) is the mass of the bullet, \(c\) is the specific heat capacity, and \(\Delta T\) is the change in temperature. For lead, the specific heat capacity is 128 J/kgK. So, \(\Delta T = 47^\circ\text{C} - 0^\circ\text{C} = 47 \text{K}\) \(Q_{bullet} = (0.020 \text{kg})\times(128 \frac{\text{J}}{\text{kg} \cdot \text{K}}) \times (47 \text{K})\)
02

Calculate the initial kinetic energy of the bullet

Next, we need to calculate the initial kinetic energy of the bullet using the formula \(KE = 0.5mv^2\), where \(KE\) is the kinetic energy, \(m\) is the mass of the bullet, and \(v\) is the velocity. \(KE_{bullet} = 0.5\times(0.020 \text{kg})\times(5.00\times10^2\frac{\text{m}}{\text{s}})^2\)
03

Calculate the total energy transferred to the ice

According to the energy conservation principle, the total energy transferred to the ice is the sum of the initial heat energy of the bullet and its initial kinetic energy. \(E_{total} = Q_{bullet} + KE_{bullet}\)
04

Determine the amount of ice that melts

Now, we need to find how much ice melts due to this energy input. Ice melts at a temperature of \(0^\circ\text{C}\), and its latent heat of fusion is \(L_f = 3.33\times10^5 \frac{\text{J}}{\text{kg}}\). Thus, the mass of ice melted can be calculated using the formula \(Q_{melted} = mL_f\), where \(Q_{melted}\) is the energy required to melt the ice and \(m\) is the mass of ice. So, \(m_{ice} = \frac{E_{total}}{L_f}\) By substituting the values calculated in steps 1-3 into the above formula, we can find the mass of ice that melts.

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