A steel ball of mass `m_(1)=1kg` moving with velocity `50ms^(-1)` collides with another ball of mass `m_(2)=200g` lying on the ground. Due the collision, the KE is lost and their internal energies change equally and `T_(1) and T_(2)` are the temperature changes of masses `m_(1) and m_(2)` respectively. If the specific heat of steel is unity and `J="4.18 J cal"^(-1)`, then

Options

A. `T_(1)=7.1^(@)C and T_(2)=1.47^(@)C`
B. `T_(1)=1.47^(@)C and T_(2)=7.1^(@)C`
C. `T_(1)=3.4^(@)C and T_(2)=17.0^(@)C`
D. `T_(1)=17.0^(@)C and T_(2)=3.4^(@)C`

Answer

To solve the problem, we need to analyze the collision between the two balls, calculate the kinetic energy lost, and then relate that to the temperature changes of the two masses due to the change in internal energy.

### Step 1: Calculate the initial kinetic energy of the steel ball (m1)
The kinetic energy (KE) of an object is given by the formula:
\[ KE = \frac{1}{2} m v^2 \]
For the steel ball (mass \( m_1 = 1 \, \text{kg} \) and velocity \( v_1 = 50 \, \text{m/s} \)):
\[ KE_1 = \frac{1}{2} \times 1 \, \text{kg} \times (50 \, \text{m/s})^2 = \frac{1}{2} \times 1 \times 2500 = 1250 \, \text{J} \]

### Step 2: Calculate the mass of the second ball (m2)
The mass of the second ball is given as \( m_2 = 200 \, \text{g} \). Converting this to kilograms:
\[ m_2 = 200 \, \text{g} = 0.2 \, \text{kg} \]

### Step 3: Set up the energy conservation equation
Since the internal energy change is equal for both masses, we can express this as:
\[ m_1 \cdot S \cdot T_1 = m_2 \cdot S \cdot T_2 \]
Where \( S \) is the specific heat (given as unity, \( S = 1 \)). Thus, the equation simplifies to:
\[ m_1 \cdot T_1 = m_2 \cdot T_2 \]

### Step 4: Substitute the known values
Substituting \( m_1 = 1 \, \text{kg} \) and \( m_2 = 0.2 \, \text{kg} \):
\[ 1 \cdot T_1 = 0.2 \cdot T_2 \]
This can be rearranged to find the ratio of temperature changes:
\[ \frac{T_1}{T_2} = \frac{0.2}{1} = \frac{1}{5} \]

### Step 5: Express T1 in terms of T2
From the ratio, we can express \( T_1 \) as:
\[ T_1 = \frac{T_2}{5} \]

### Step 6: Determine the total kinetic energy lost
Since the collision is inelastic, some kinetic energy is lost. The lost kinetic energy will be converted into internal energy, which causes the temperature changes. The lost kinetic energy can be calculated, but since we are only interested in the ratio of temperature changes, we can use the relationship derived above.

### Step 7: Conclusion
From the derived relationship \( T_1 = \frac{T_2}{5} \), we can conclude that the temperature change of the first ball is one-fifth of that of the second ball.