A particle movig with a uniformacceleration travels ` 24 ` metre and ` 64` metre in first two successive intervals of ` 4` seconds each. Its initial velocity is.

Options

A. ` 1 m //sec`
B. ` 10 m // sec`
C. ` 5m//sec`
D. ` 2`

Explanation

To solve the problem step by step, we will use the equations of motion for a particle moving with uniform acceleration.

### Step 1: Understand the problem
We know that a particle travels 24 meters in the first 4 seconds and then 64 meters in the next 4 seconds. We need to find the initial velocity (u) of the particle.

### Step 2: Use the first equation of motion
The distance traveled by a particle under uniform acceleration can be expressed as:
\[ s = ut + \frac{1}{2} a t^2 \]
For the first interval (0 to 4 seconds):
- Distance (s) = 24 m
- Time (t) = 4 s

Substituting these values into the equation:
\[ 24 = u(4) + \frac{1}{2} a (4^2) \]
This simplifies to:
\[ 24 = 4u + 8a \quad \text{(Equation 1)} \]

### Step 3: Use the second equation of motion
For the second interval (4 to 8 seconds), the total time is 8 seconds. The distance traveled in this interval is 64 m. The initial velocity for this interval becomes the final velocity from the first interval:
\[ v = u + at \]
At t = 4 s, we have:
\[ v = u + 4a \]
Now, using the same distance formula for the second interval:
\[ 64 = v(4) + \frac{1}{2} a (4^2) \]
Substituting \(v\) from the previous equation:
\[ 64 = (u + 4a)(4) + 8a \]
This simplifies to:
\[ 64 = 4u + 16a + 8a \]
Thus:
\[ 64 = 4u + 24a \quad \text{(Equation 2)} \]

### Step 4: Solve the equations simultaneously
We now have two equations:
1. \( 4u + 8a = 24 \) (Equation 1)
2. \( 4u + 24a = 64 \) (Equation 2)

We can subtract Equation 1 from Equation 2:
\[ (4u + 24a) - (4u + 8a) = 64 - 24 \]
This simplifies to:
\[ 16a = 40 \]
Thus:
\[ a = \frac{40}{16} = 2.5 \, \text{m/s}^2 \]

### Step 5: Substitute back to find initial velocity
Now substitute the value of \(a\) back into Equation 1:
\[ 4u + 8(2.5) = 24 \]
This leads to:
\[ 4u + 20 = 24 \]
So:
\[ 4u = 4 \]
Thus:
\[ u = 1 \, \text{m/s} \]

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

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