A body of mass m collides elastically with another body at rest and then continues to move in the original continues to move in the original direction with one half of its original speed. mass of the body is

Options

A. `m`
B. `2/3 m`
C. `m/3`
D. `m/2`

Answer

To solve the problem, we will use the principles of conservation of momentum and the properties of elastic collisions. Here’s a step-by-step breakdown of the solution:

### Step 1: Define the Variables
Let:
- Mass of the first body = \( m \)
- Initial velocity of the first body = \( V \)
- Mass of the second body = \( M \) (unknown)
- Initial velocity of the second body = \( 0 \) (at rest)
- Final velocity of the first body after collision = \( \frac{V}{2} \)
- Final velocity of the second body after collision = \( V_2 \) (unknown)

### Step 2: Apply Conservation of Momentum
According to the law of conservation of momentum:
\[ \text{Initial Momentum} = \text{Final Momentum} \]
The initial momentum is:
\[ mV + 0 = mV \]
The final momentum is:
\[ m \left(\frac{V}{2}\right) + MV_2 \]
Setting the initial momentum equal to the final momentum gives us:
\[ mV = m \left(\frac{V}{2}\right) + MV_2 \]

### Step 3: Simplify the Momentum Equation
Rearranging the equation:
\[ mV - m\left(\frac{V}{2}\right) = MV_2 \]
This simplifies to:
\[ mV - \frac{mV}{2} = MV_2 \]
\[ \frac{mV}{2} = MV_2 \tag{1} \]

### Step 4: Use the Coefficient of Restitution
For an elastic collision, the coefficient of restitution \( e = 1 \). The formula for the coefficient of restitution is:
\[ e = \frac{\text{Relative velocity after collision}}{\text{Relative velocity before collision}} \]
Thus:
\[ V_2 - \frac{V}{2} = 1 \cdot \left(V - 0\right) \]
This simplifies to:
\[ V_2 - \frac{V}{2} = V \]
Rearranging gives:
\[ V_2 = V + \frac{V}{2} = \frac{3V}{2} \tag{2} \]

### Step 5: Substitute \( V_2 \) Back into the Momentum Equation
Substituting equation (2) into equation (1):
\[ \frac{mV}{2} = M \left(\frac{3V}{2}\right) \]
Cancelling \( V \) from both sides (assuming \( V \neq 0 \)):
\[ \frac{m}{2} = \frac{3M}{2} \]
Multiplying both sides by 2:
\[ m = 3M \]
Thus, we can express \( M \) in terms of \( m \):
\[ M = \frac{m}{3} \]

### Conclusion
The mass of the second body is:
\[ \boxed{\frac{m}{3}} \]