Take the mechanical equivalent of heat as 4J`//`cal. A 10-gram bullet moving at 2000 `m//s` plunges into 1 kg of paraffin wax ( specific heat 0.7 cal `// ( g^(@) C )`. The wax was initially at `20^(@)C`. Assuming that all the bullet's energy heats the wax, its final temperature is

Options

A. `20.14^(@)C`
B. `23.5^(@)C`
C. `20.006^(@)C`
D. `27.1^(@)C`

Answer

To solve the problem step by step, we will follow these calculations:

### Step 1: Calculate the Kinetic Energy of the Bullet
The kinetic energy (KE) of the bullet can be calculated using the formula:
\[ KE = \frac{1}{2} mv^2 \]
where:
- \( m \) is the mass of the bullet in kilograms (10 grams = 0.01 kg),
- \( v \) is the velocity of the bullet (2000 m/s).

Substituting the values:
\[ KE = \frac{1}{2} \times 0.01 \, \text{kg} \times (2000 \, \text{m/s})^2 \]
\[ KE = \frac{1}{2} \times 0.01 \times 4000000 \]
\[ KE = 0.005 \times 4000000 = 20000 \, \text{J} \]

### Step 2: Convert Kinetic Energy from Joules to Calories
Given that the mechanical equivalent of heat is \( 4 \, \text{J/cal} \), we can convert the energy from joules to calories:
\[ \text{Energy in calories} = \frac{20000 \, \text{J}}{4 \, \text{J/cal}} = 5000 \, \text{cal} \]

### Step 3: Calculate the Heat Absorbed by the Paraffin Wax
The heat absorbed by the paraffin wax can be calculated using the formula:
\[ Q = mc\Delta T \]
where:
- \( Q \) is the heat absorbed (5000 cal),
- \( m \) is the mass of the wax (1 kg = 1000 g),
- \( c \) is the specific heat of the wax (0.7 cal/g°C),
- \( \Delta T \) is the change in temperature.

Rearranging the formula to find \( \Delta T \):
\[ \Delta T = \frac{Q}{mc} \]
Substituting the values:
\[ \Delta T = \frac{5000 \, \text{cal}}{1000 \, \text{g} \times 0.7 \, \text{cal/g°C}} = \frac{5000}{700} \approx 7.14 \, °C \]

### Step 4: Calculate the Final Temperature of the Wax
The final temperature \( T_f \) of the wax can be calculated by adding the change in temperature \( \Delta T \) to the initial temperature \( T_i \):
\[ T_f = T_i + \Delta T \]
Given that the initial temperature \( T_i \) is \( 20°C \):
\[ T_f = 20°C + 7.14°C = 27.14°C \]

### Final Answer
The final temperature of the paraffin wax is approximately \( 27.14°C \).

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