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

Options

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

Explanation

To solve the problem step by step, we will use the equations of motion under uniform acceleration.

### Step 1: Define Variables
Let:
- \( U \) = initial velocity (m/s)
- \( A \) = acceleration (m/s²)

### Step 2: Analyze the First 3 Seconds
The distance covered in the first 3 seconds is given as 12 m. We can use the equation of motion:
\[ S = Ut + \frac{1}{2} A t^2 \]
For the first 3 seconds (\( t = 3 \) s):
\[ 12 = U(3) + \frac{1}{2} A (3^2) \]
This simplifies to:
\[ 12 = 3U + \frac{9}{2} A \]
Rearranging gives us:
\[ 3U + \frac{9}{2} A = 12 \quad \text{(Equation 1)} \]

### Step 3: Analyze the Next 3 Seconds
The distance covered in the next 3 seconds (from \( t = 3 \) s to \( t = 6 \) s) is 30 m. The total distance covered from \( t = 0 \) to \( t = 6 \) seconds is:
\[ 12 + 30 = 42 \text{ m} \]
Using the same equation of motion for \( t = 6 \) s:
\[ 42 = U(6) + \frac{1}{2} A (6^2) \]
This simplifies to:
\[ 42 = 6U + \frac{1}{2} A (36) \]
Which further simplifies to:
\[ 42 = 6U + 18A \]
Rearranging gives us:
\[ 6U + 18A = 42 \quad \text{(Equation 2)} \]

### Step 4: Solve the System of Equations
Now we have two equations:
1. \( 3U + \frac{9}{2} A = 12 \)
2. \( 6U + 18A = 42 \)

Let's simplify Equation 1:
Multiply the entire equation by 2 to eliminate the fraction:
\[ 6U + 9A = 24 \quad \text{(Equation 3)} \]

Now we can subtract Equation 2 from Equation 3:
\[ (6U + 9A) - (6U + 18A) = 24 - 42 \]
This simplifies to:
\[ -9A = -18 \]
Thus, we find:
\[ A = 2 \text{ m/s}^2 \]

### Step 5: Substitute Back to Find Initial Velocity
Now substitute \( A = 2 \) into Equation 1:
\[ 3U + \frac{9}{2}(2) = 12 \]
This simplifies to:
\[ 3U + 9 = 12 \]
Rearranging gives:
\[ 3U = 3 \]
Thus:
\[ U = 1 \text{ m/s} \]

### Final Answer
The initial velocity of the particle is \( 1 \text{ m/s} \).

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