A body of mass 1 kg moving with velocity 5 m/s collides with another body of mass 2 kg moving with velocity 1.5 m/s in opposite direction. If the coefficient of restitution is 0.8, their velocities aftter collision respectively are

Options

A. `-2.8" m"//"s",+2.4" m"//"s"`
B. `+2.8" m"//"s",-2.4" m"//"s"`
C. `-1.67" m"//"s",-2.4" m"//"s"`
D. `+2.8" m"//"s",-6.84" m"//"s"`

Answer

To solve the problem step by step, we will follow the principles of the coefficient of restitution and conservation of momentum.

### Step 1: Understand the Given Data
- Mass of body 1 (m1) = 1 kg
- Initial velocity of body 1 (u1) = 5 m/s (moving in the positive direction)
- Mass of body 2 (m2) = 2 kg
- Initial velocity of body 2 (u2) = -1.5 m/s (moving in the opposite direction)

### Step 2: Coefficient of Restitution
The coefficient of restitution (e) is given as 0.8. The formula for the coefficient of restitution is:
\[ e = \frac{\text{Velocity of separation}}{\text{Velocity of approach}} \]

### Step 3: Calculate Velocity of Approach
The velocity of approach is the sum of the magnitudes of the velocities of the two bodies before the collision:
\[ \text{Velocity of approach} = u1 + |u2| = 5 + 1.5 = 6.5 \, \text{m/s} \]

### Step 4: Set Up the Equation for Velocity of Separation
Using the coefficient of restitution:
\[ e = \frac{v2 - v1}{u1 + |u2|} \]
Substituting the known values:
\[ 0.8 = \frac{v2 - v1}{6.5} \]
From this, we can express the velocity of separation:
\[ v2 - v1 = 0.8 \times 6.5 = 5.2 \]
Thus, we have our first equation:
\[ v2 - v1 = 5.2 \quad \text{(1)} \]

### Step 5: Apply Conservation of Momentum
The conservation of momentum states:
\[ m1 \cdot u1 + m2 \cdot u2 = m1 \cdot v1 + m2 \cdot v2 \]
Substituting the values:
\[ 1 \cdot 5 + 2 \cdot (-1.5) = 1 \cdot v1 + 2 \cdot v2 \]
This simplifies to:
\[ 5 - 3 = v1 + 2v2 \]
Thus, we have our second equation:
\[ 2 = v1 + 2v2 \quad \text{(2)} \]

### Step 6: Solve the Equations Simultaneously
From equation (1):
\[ v2 = v1 + 5.2 \]
Substituting this into equation (2):
\[ 2 = v1 + 2(v1 + 5.2) \]
Expanding this gives:
\[ 2 = v1 + 2v1 + 10.4 \]
Combining like terms:
\[ 2 = 3v1 + 10.4 \]
Rearranging gives:
\[ 3v1 = 2 - 10.4 \]
\[ 3v1 = -8.4 \]
Thus:
\[ v1 = -2.8 \, \text{m/s} \]

### Step 7: Find v2
Now, substituting \( v1 \) back into equation (1):
\[ v2 = -2.8 + 5.2 = 2.4 \, \text{m/s} \]

### Final Answer
The velocities after the collision are:
- Velocity of body 1 (v1) = -2.8 m/s
- Velocity of body 2 (v2) = 2.4 m/s