A particle starting with certain initial velocity and uniform acceleration covers a distance of 12 m in first 3 seconds and a distance of 30 m in next 3 seconds. The initial velocity of the particle is

Options

A. `1 ms ^(-1)`
B. `2 ms ^(-1)`
C. `2.5 ms ^(-1)`
D. `3 ms ^(-1)`

Explanation

To solve the problem step by step, we will use the equations of motion to find the initial velocity of the particle.

### Step 1: Understand the problem
The particle covers a distance of 12 m in the first 3 seconds and 30 m in the next 3 seconds. We need to find the initial velocity (u) of the particle and we assume uniform acceleration (a).

### Step 2: Set up the equations
We will use the second equation of motion:
\[ s = ut + \frac{1}{2} a t^2 \]

### Step 3: Write the first equation for the first 3 seconds
For the first 3 seconds (t = 3 s), the distance covered (s) is 12 m:
\[ 12 = u(3) + \frac{1}{2} a (3^2) \]
This simplifies to:
\[ 12 = 3u + \frac{9}{2} a \]
Multiply through by 2 to eliminate the fraction:
\[ 24 = 6u + 9a \]
This is our **Equation 1**.

### Step 4: Write the second equation for the first 6 seconds
Now, for the total time of 6 seconds (t = 6 s), the total distance covered is 12 m + 30 m = 42 m:
\[ 42 = u(6) + \frac{1}{2} a (6^2) \]
This simplifies to:
\[ 42 = 6u + 18a \]
Multiply through by 2:
\[ 84 = 12u + 36a \]
This is our **Equation 2**.

### Step 5: Solve the equations simultaneously
We have the two equations:
1. \( 24 = 6u + 9a \) (Equation 1)
2. \( 84 = 12u + 36a \) (Equation 2)

We can simplify Equation 2 by dividing everything by 12:
\[ 7 = u + 3a \]
This is our **Equation 3**.

### Step 6: Express 'a' in terms of 'u'
From Equation 1:
\[ 6u + 9a = 24 \]
Rearranging gives:
\[ 9a = 24 - 6u \]
\[ a = \frac{24 - 6u}{9} \]

### Step 7: Substitute 'a' into Equation 3
Now substitute this expression for 'a' into Equation 3:
\[ 7 = u + 3\left(\frac{24 - 6u}{9}\right) \]
Multiply through by 9 to eliminate the fraction:
\[ 63 = 9u + 3(24 - 6u) \]
Expanding gives:
\[ 63 = 9u + 72 - 18u \]
Combine like terms:
\[ 63 = -9u + 72 \]
Rearranging gives:
\[ 9u = 72 - 63 \]
\[ 9u = 9 \]
\[ u = 1 \text{ m/s} \]

### Final Answer
The initial velocity of the particle is **1 m/s**.

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