A body starting from rest, accelerates at a constant rate a `m//s^2` for some time after which it decelerates at a constant rate b `m//s^2` to come to rest finally. If the total time elapsed is t sec, the maximum velocity attained by the body is given by

Options

A. `(ab)/(a+b) tm//s`
B. `(ab)/(a-b) tm//s`
C. `(2ab)/(a+b) tm//s`
D. `(2ab)/(a-b) tm//s`

Explanation

To solve the problem of finding the maximum velocity attained by a body that starts from rest, accelerates at a constant rate \( a \, \text{m/s}^2 \) for some time, and then decelerates at a constant rate \( b \, \text{m/s}^2 \) to come to rest, we can break down the solution into several steps.

### Step-by-Step Solution:

1. **Define Variables**:
- Let \( T_1 \) be the time during which the body accelerates.
- Let \( T_2 \) be the time during which the body decelerates.
- The total time \( T \) is given by:
\[ T = T_1 + T_2 \]

2. **Use the First Equation of Motion for Acceleration**:
- Since the body starts from rest, the initial velocity \( u = 0 \).
- The final velocity after accelerating for time \( T_1 \) is given by:
\[ V_{\text{max}} = u + a T_1 = 0 + a T_1 = a T_1 \]
- This is our **Equation 1**.

3. **Use the First Equation of Motion for Deceleration**:
- The body comes to rest after decelerating, so the final velocity \( v = 0 \).
- The initial velocity for this phase is \( V_{\text{max}} \).
- The equation for deceleration gives:
\[ 0 = V_{\text{max}} - b T_2 \implies V_{\text{max}} = b T_2 \]
- This is our **Equation 2**.

4. **Relate \( T_1 \) and \( T_2 \)**:
- From Equation 1 and Equation 2, we can equate \( V_{\text{max}} \):
\[ a T_1 = b T_2 \]
- Rearranging gives:
\[ T_1 = \frac{b T_2}{a} \]

5. **Substitute \( T_1 \) in Total Time Equation**:
- Substitute \( T_1 \) into the total time equation:
\[ T = T_1 + T_2 = \frac{b T_2}{a} + T_2 \]
- Factoring out \( T_2 \):
\[ T = T_2 \left( \frac{b}{a} + 1 \right) = T_2 \left( \frac{b + a}{a} \right) \]
- Solving for \( T_2 \):
\[ T_2 = \frac{T a}{a + b} \]

6. **Substitute \( T_2 \) back to find \( V_{\text{max}} \)**:
- Substitute \( T_2 \) into Equation 2:
\[ V_{\text{max}} = b T_2 = b \left( \frac{T a}{a + b} \right) = \frac{ab T}{a + b} \]

### Final Result:
The maximum velocity attained by the body is given by:
\[ V_{\text{max}} = \frac{ab T}{a + b} \]