A body of mass m, moving with velocity `mu` collides elasticity with another body at rest having mass M. If the body of mass M moves with velocity V, then the velocity of the body of mass m after the impact is

Options

A. `(m-M)/(m + M)mu`
B. `(m-M)/(m + M)V`
C. `(m + MV)/(m + M)`
D. `("m"u-MV)/m`

Answer

To solve the problem step by step, we will use the principle of conservation of linear momentum. Here's how to approach it:

### Step 1: Understand the scenario
We have two bodies:
- Body 1 (mass = m) moving with initial velocity = u.
- Body 2 (mass = M) at rest (initial velocity = 0).

After the collision, body 2 moves with velocity = V, and we need to find the final velocity of body 1 (let's denote it as x).

### Step 2: Write the conservation of momentum equation
According to the conservation of linear momentum, the total momentum before the collision is equal to the total momentum after the collision.

**Initial momentum**:
- Momentum of body 1 = m * u
- Momentum of body 2 = M * 0 = 0 (since it is at rest)

So, the total initial momentum = m * u + 0 = m * u.

**Final momentum**:
- After the collision, the momentum of body 1 = m * x (where x is the final velocity of body 1).
- The momentum of body 2 = M * V.

Thus, the total final momentum = m * x + M * V.

### Step 3: Set up the equation
Setting the initial momentum equal to the final momentum, we have:
\[ m * u = m * x + M * V \]

### Step 4: Solve for x
Rearranging the equation to solve for x:
\[ m * x = m * u - M * V \]
\[ x = \frac{m * u - M * V}{m} \]

### Step 5: Simplify the expression
We can express x in a more usable form:
\[ x = u - \frac{M * V}{m} \]

### Conclusion
Thus, the final velocity of the body of mass m after the impact is:
\[ x = u - \frac{M * V}{m} \]

### Answer
The velocity of the body of mass m after the impact is:
\[ x = \frac{m * u - M * V}{m} \]