Two particles P and Q simultaneously start moving from point A with velocities `15 m//s` and `20 m//s` respectively. The two particles move with acceleration equal in magnitude but opposite in direction. When P overtakes Q and B then its velocity is `30 m//s` the velocity of Q at point B will be

Options

A. `30 m//s`
B. `5 m//s`
C. `10 m//s`
D. `15 m//s`

Explanation

To solve the problem, we will analyze the motion of both particles P and Q step by step.

### Step 1: Understand the Initial Conditions
- Particle P starts with an initial velocity \( u_P = 15 \, \text{m/s} \).
- Particle Q starts with an initial velocity \( u_Q = 20 \, \text{m/s} \).
- Both particles start from the same point A at the same time.

**Hint:** Identify the initial velocities of both particles and note that they start from the same position.

### Step 2: Define the Accelerations
- Let the acceleration of particle P be \( a \) (positive).
- Since the accelerations are equal in magnitude but opposite in direction, the acceleration of particle Q will be \( -a \) (negative).

**Hint:** Remember that the direction of acceleration affects the velocity change over time.

### Step 3: Use the Velocity Formula for Particle P
- When particle P overtakes particle Q at point B, its final velocity is given as \( v_P = 30 \, \text{m/s} \).
- We can use the equation of motion:
\[ v = u + at \]
For particle P:
\[ 30 = 15 + at \]
Rearranging gives:
\[ at = 30 - 15 = 15 \quad \text{(1)} \]

**Hint:** Use the equation of motion to relate final velocity, initial velocity, acceleration, and time.

### Step 4: Use the Velocity Formula for Particle Q
- Let the final velocity of particle Q at point B be \( v_Q \).
- For particle Q, we can use the same equation of motion:
\[ v_Q = u_Q + (-a)t \]
Substituting the known values:
\[ v_Q = 20 - at \quad \text{(2)} \]

**Hint:** Set up the equation for particle Q similarly, noting that its acceleration is negative.

### Step 5: Substitute the Value of \( at \) from Equation (1) into Equation (2)
- From Equation (1), we have \( at = 15 \).
- Substitute this into Equation (2):
\[ v_Q = 20 - 15 \]
Simplifying gives:
\[ v_Q = 5 \, \text{m/s} \]

**Hint:** Substitute the known value of \( at \) to find the final velocity of particle Q.

### Conclusion
The velocity of particle Q at point B is \( 5 \, \text{m/s} \).

**Final Answer:** The velocity of Q at point B will be \( 5 \, \text{m/s} \).