Two elastic bodies P and Q having equal mases are moving along the same line with velocities of 16 m/s and 10 m/s respectively. Their velocities after the elastic collision will be in m/s :

Options

A. 0 and 25
B. 5 and 20
C. 10 and 16
D. 20 and 5

Answer

To solve the problem of finding the velocities of two elastic bodies P and Q after an elastic collision, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Data:**
- Mass of body P (m1) = m
- Mass of body Q (m2) = m
- Initial velocity of body P (u1) = 16 m/s
- Initial velocity of body Q (u2) = 10 m/s

2. **Conservation of Momentum:**
In an elastic collision, the total momentum before the collision is equal to the total momentum after the collision. Therefore, we can write:
\[ m \cdot u1 + m \cdot u2 = m \cdot v1 + m \cdot v2 \]
Since the masses are equal, we can simplify this to:
\[ u1 + u2 = v1 + v2 \]
Substituting the values:
\[ 16 + 10 = v1 + v2 \]
\[ 26 = v1 + v2 \quad \text{(Equation 1)} \]

3. **Relative Velocity in Elastic Collisions:**
For elastic collisions, the relative velocity of separation is equal to the relative velocity of approach. This can be expressed as:
\[ v2 - v1 = -(u2 - u1) \]
Substituting the values:
\[ v2 - v1 = -(10 - 16) \]
\[ v2 - v1 = 6 \quad \text{(Equation 2)} \]

4. **Solving the Equations:**
Now we have two equations:
- Equation 1: \( v1 + v2 = 26 \)
- Equation 2: \( v2 - v1 = 6 \)

From Equation 2, we can express \( v2 \) in terms of \( v1 \):
\[ v2 = v1 + 6 \]

Substitute this expression for \( v2 \) into Equation 1:
\[ v1 + (v1 + 6) = 26 \]
\[ 2v1 + 6 = 26 \]
\[ 2v1 = 20 \]
\[ v1 = 10 \text{ m/s} \]

5. **Finding \( v2 \):**
Now substitute \( v1 \) back into the expression for \( v2 \):
\[ v2 = v1 + 6 = 10 + 6 = 16 \text{ m/s} \]

### Final Result:
The velocities after the elastic collision are:
- Velocity of body P (v1) = 10 m/s
- Velocity of body Q (v2) = 16 m/s