Question

A metallic bullet with an initial velocity of 500 m/s penetrates a solid object and melts. The initial temperature of the bullet is 30°C and its melting point is 280°C. The ratio of total heat generated to the initial kinetic energy of the bullet will be : [Latent heat of fusion of metal=3.0x104 J/kg and specific heat capacity of metal=200 J/kg-K]

• A. \( 0.5 \)
• B. \( 0.64 \)
• C. \( 0.81 \)
• D. \( 0.11 \)
• E. \( 0.34 \)

Answer 1

The Correct Option is B

Given parameters:

• Initial velocity \( v = 500 \, \text{m/s} \)
• Initial temperature \( T_i = 30^\circ \text{C} \)
• Melting point \( T_m = 280^\circ \text{C} \)
• Specific heat \( c = 200 \, \text{J/kg-K} \)
• Latent heat \( L = 3.0 \times 10^4 \, \text{J/kg} \)

 

Initial kinetic energy: \[ K = \frac{1}{2}mv^2 \] \[ K = \frac{1}{2}m(500)^2 = 125000m \, \text{J} \]

Heat required for temperature rise: \[ Q_1 = mc\Delta T = m \times 200 \times (280-30) = 50000m \, \text{J} \]

Heat required for melting: \[ Q_2 = mL = m \times 3.0 \times 10^4 = 30000m \, \text{J} \]

Total heat generated: \[ Q_{total} = Q_1 + Q_2 = 80000m \, \text{J} \]

Ratio calculation: \[ \frac{Q_{total}}{K} = \frac{80000m}{125000m} = 0.64 \]

Thus, the correct option is (B): \( 0.64 \).

Answer 2

1. Calculate the initial kinetic energy (KE):

Let m be the mass of the bullet. The initial kinetic energy is given by:

\[KE = \frac{1}{2}mv^2 = \frac{1}{2}m(500^2) = 125000m \, J\]

2. Calculate the heat required to raise the temperature to the melting point (Q1):

\[Q_1 = mc\Delta T\]

where:

• m is the mass
• c = 200 J/kg-K (specific heat capacity)
• ΔT = 280°C - 30°C = 250 K (change in temperature)

\[Q_1 = m(200)(250) = 50000m \, J\]

3. Calculate the heat required for melting (Q2):

\[Q_2 = mL\]

where L = 3.0 × 10⁴ J/kg (latent heat of fusion)

\[Q_2 = m(3.0 \times 10^4) = 30000m \, J\]

4. Calculate the total heat generated (Q):

The total heat generated is the sum of the heat required to raise the temperature and the heat required for melting:

\[Q = Q_1 + Q_2 = 50000m + 30000m = 80000m \, J\]

5. Calculate the ratio of total heat generated to initial kinetic energy:

\[\frac{Q}{KE} = \frac{80000m}{125000m} = \frac{80}{125} = \frac{16}{25} = 0.64\]