An object of mass 80 kg moving with velocity `2ms^(-1)` hit by collides with another object of mass 20 kg moving with velocity `4 ms ^(-1)` Find the loss of energy assuming a perfectly, inelastic collision

Options

A. 12 J
B. 24 J
C. 30 J
D. 32 J

Answer

To solve the problem of finding the loss of energy during a perfectly inelastic collision between two objects, we can follow these steps:

### Step 1: Identify the masses and initial velocities
Let:
- Mass of object 1, \( m_1 = 80 \, \text{kg} \)
- Initial velocity of object 1, \( u_1 = 2 \, \text{m/s} \)
- Mass of object 2, \( m_2 = 20 \, \text{kg} \)
- Initial velocity of object 2, \( u_2 = 4 \, \text{m/s} \)

### Step 2: Apply the conservation of momentum
In a perfectly inelastic collision, the total momentum before the collision is equal to the total momentum after the collision. The formula for conservation of momentum is:
\[ m_1 u_1 + m_2 u_2 = (m_1 + m_2) v \]
where \( v \) is the final velocity after the collision.

### Step 3: Substitute the values into the momentum equation
Substituting the known values:
\[ 80 \times 2 + 20 \times 4 = (80 + 20) v \]
Calculating the left side:
\[ 160 + 80 = 100v \]
This simplifies to:
\[ 240 = 100v \]

### Step 4: Solve for the final velocity \( v \)
\[ v = \frac{240}{100} = 2.4 \, \text{m/s} \]

### Step 5: Calculate the initial kinetic energy
The initial kinetic energy (KE) of the system before the collision is given by:
\[ KE_1 = \frac{1}{2} m_1 u_1^2 + \frac{1}{2} m_2 u_2^2 \]
Substituting the values:
\[ KE_1 = \frac{1}{2} \times 80 \times (2)^2 + \frac{1}{2} \times 20 \times (4)^2 \]
Calculating each term:
\[ = \frac{1}{2} \times 80 \times 4 + \frac{1}{2} \times 20 \times 16 \]
\[ = 160 + 160 = 320 \, \text{J} \]

### Step 6: Calculate the final kinetic energy
The final kinetic energy after the collision is:
\[ KE_2 = \frac{1}{2} (m_1 + m_2) v^2 \]
Substituting the values:
\[ KE_2 = \frac{1}{2} \times (80 + 20) \times (2.4)^2 \]
Calculating:
\[ = \frac{1}{2} \times 100 \times 5.76 \]
\[ = 50 \times 5.76 = 288 \, \text{J} \]

### Step 7: Calculate the loss of kinetic energy
The loss of kinetic energy is given by:
\[ \text{Loss of KE} = KE_1 - KE_2 \]
Substituting the values:
\[ \text{Loss of KE} = 320 \, \text{J} - 288 \, \text{J} = 32 \, \text{J} \]

### Final Answer:
The loss of energy during the collision is **32 Joules**.

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