A particles starts from rest and has an acceleration of ` 2m//s^(2)` for 10 sec. After that , it travels for 30 sec with constant speed and then undergoes a retardation of ` 4m//s^(2)` and comes back to rest. The total distance covered by the particle is

Options

A. 650 m
B. 750 m
C. 700 m
D. 800 m

Explanation

To solve the problem step by step, we will break down the motion of the particle into three phases:

1. **Acceleration Phase (0 to 10 seconds)**
2. **Constant Speed Phase (10 to 40 seconds)**
3. **Retardation Phase (40 seconds to rest)**

### Step 1: Calculate the distance during the acceleration phase (0 to 10 seconds)

- **Initial velocity (u)** = 0 m/s (starts from rest)
- **Acceleration (a)** = 2 m/s²
- **Time (t)** = 10 s

Using the equation of motion:
\[ s_1 = ut + \frac{1}{2} a t^2 \]

Substituting the values:
\[ s_1 = 0 \cdot 10 + \frac{1}{2} \cdot 2 \cdot (10)^2 \]
\[ s_1 = 0 + \frac{1}{2} \cdot 2 \cdot 100 \]
\[ s_1 = 100 \, \text{meters} \]

### Step 2: Calculate the distance during the constant speed phase (10 to 40 seconds)

- **Final velocity after acceleration (v)** can be calculated using:
\[ v = u + at \]
\[ v = 0 + 2 \cdot 10 = 20 \, \text{m/s} \]

- **Time during constant speed** = 30 seconds

The distance covered during this phase:
\[ s_2 = v \cdot t \]
\[ s_2 = 20 \cdot 30 = 600 \, \text{meters} \]

### Step 3: Calculate the distance during the retardation phase (40 seconds to rest)

- **Initial velocity (u)** = 20 m/s (from the end of the constant speed phase)
- **Final velocity (v)** = 0 m/s (comes to rest)
- **Retardation (a)** = -4 m/s²

Using the equation of motion:
\[ v^2 = u^2 + 2as \]
Substituting the values:
\[ 0 = (20)^2 + 2 \cdot (-4) \cdot s_3 \]
\[ 0 = 400 - 8s_3 \]
\[ 8s_3 = 400 \]
\[ s_3 = \frac{400}{8} = 50 \, \text{meters} \]

### Step 4: Calculate the total distance covered by the particle

Now, we can find the total distance \( S \) covered by the particle:
\[ S = s_1 + s_2 + s_3 \]
\[ S = 100 + 600 + 50 \]
\[ S = 750 \, \text{meters} \]

### Final Answer
The total distance covered by the particle is **750 meters**.

---