A body starts from rest with an acceleration `2m//s^(2)` till it attains the maximum velocity then retards to rest with `3m//s^(2)`. If total time taken is 10 second, then maximum speed attained is

Options

A. 12 m/s
B. 8 m/s
C. 6 m/s
D. 4 m/s

Answer

To solve the problem step by step, we will break it down into parts based on the motion of the body.

### Step 1: Understanding the Motion
The body has two phases of motion:
1. **Acceleration Phase**: The body accelerates from rest with an acceleration of \(2 \, \text{m/s}^2\) until it reaches its maximum velocity \(v_{\text{max}}\).
2. **Retardation Phase**: The body then decelerates (retards) to rest with a retardation of \(3 \, \text{m/s}^2\).

### Step 2: Setting Up the Equations
Let:
- \(t_1\) = time taken to reach maximum velocity
- \(t_2\) = time taken to come to rest

From the problem, we know that the total time taken is:
\[ t_1 + t_2 = 10 \, \text{s} \]

### Step 3: Finding Time for Each Phase
Using the equations of motion, we can express \(t_1\) and \(t_2\) in terms of \(v_{\text{max}}\):

1. **For the acceleration phase**:
\[ v_{\text{max}} = u + at_1 \]
Here, \(u = 0\) (initial velocity), \(a = 2 \, \text{m/s}^2\), thus:
\[ v_{\text{max}} = 0 + 2t_1 \implies t_1 = \frac{v_{\text{max}}}{2} \]

2. **For the retardation phase**:
\[ 0 = v_{\text{max}} - 3t_2 \]
Rearranging gives:
\[ 3t_2 = v_{\text{max}} \implies t_2 = \frac{v_{\text{max}}}{3} \]

### Step 4: Substitute into Total Time Equation
Now substitute \(t_1\) and \(t_2\) into the total time equation:
\[ \frac{v_{\text{max}}}{2} + \frac{v_{\text{max}}}{3} = 10 \]

### Step 5: Finding a Common Denominator
The common denominator for 2 and 3 is 6. Thus, we can rewrite the equation:
\[ \frac{3v_{\text{max}}}{6} + \frac{2v_{\text{max}}}{6} = 10 \]
Combining the fractions gives:
\[ \frac{5v_{\text{max}}}{6} = 10 \]

### Step 6: Solve for \(v_{\text{max}}\)
Now, multiply both sides by 6:
\[ 5v_{\text{max}} = 60 \]
Dividing by 5:
\[ v_{\text{max}} = 12 \, \text{m/s} \]

### Conclusion
The maximum speed attained by the body is:
\[ \boxed{12 \, \text{m/s}} \]