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 distinct phases: acceleration, constant speed, and retardation.

### Step 1: Calculate the distance during acceleration (s1)

The particle starts from rest (initial velocity, u = 0) and accelerates at \( a = 2 \, \text{m/s}^2 \) for \( t = 10 \, \text{s} \).

Using the formula for distance under constant acceleration:
\[ s_1 = ut + \frac{1}{2} a t^2 \]
Substituting the values:
\[ s_1 = 0 \cdot 10 + \frac{1}{2} \cdot 2 \cdot (10)^2 \]
\[ s_1 = 0 + \frac{1}{2} \cdot 2 \cdot 100 \]
\[ s_1 = 100 \, \text{m} \]

### Step 2: Calculate the speed at the end of acceleration

To find the speed at the end of the acceleration phase (v1), we use the formula:
\[ v = u + at \]
Substituting the values:
\[ v_1 = 0 + 2 \cdot 10 \]
\[ v_1 = 20 \, \text{m/s} \]

### Step 3: Calculate the distance during constant speed (s2)

The particle travels at a constant speed of \( 20 \, \text{m/s} \) for \( 30 \, \text{s} \).

Using the formula for distance:
\[ s_2 = v \cdot t \]
Substituting the values:
\[ s_2 = 20 \cdot 30 \]
\[ s_2 = 600 \, \text{m} \]

### Step 4: Calculate the distance during retardation (s3)

The particle then undergoes a retardation of \( a = -4 \, \text{m/s}^2 \) until it comes to rest (final velocity, v = 0).

Using the formula:
\[ v^2 = u^2 + 2as \]
Rearranging for s:
\[ 0 = (20)^2 + 2 \cdot (-4) \cdot s_3 \]
\[ 0 = 400 - 8s_3 \]
\[ 8s_3 = 400 \]
\[ s_3 = \frac{400}{8} = 50 \, \text{m} \]

### Step 5: Calculate the total distance covered

Now, we can find the total distance covered by the particle:
\[ s_{\text{total}} = s_1 + s_2 + s_3 \]
Substituting the values:
\[ s_{\text{total}} = 100 + 600 + 50 \]
\[ s_{\text{total}} = 750 \, \text{m} \]

### Final Answer

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

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