A body is moving with uniform acceleration describes 40 m in the first 5 sec and 65 m in next 5 sec. Its initial velocity will be

Options

A. 4 m/s
B. 2.5 m/s
C. 5.5 m/s
D. 11 m/s

Explanation

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

### Step 1: Define the Variables
Let:
- \( u \) = initial velocity (m/s)
- \( a \) = acceleration (m/s²)
- \( t_1 = 5 \) seconds (first interval)
- \( t_2 = 5 \) seconds (second interval)
- Total time \( t = t_1 + t_2 = 10 \) seconds

### Step 2: Write the Equation for the First 5 Seconds
The distance covered in the first 5 seconds is given as 40 m. Using the equation of motion:
\[ s_1 = ut_1 + \frac{1}{2} a t_1^2 \]
Substituting the known values:
\[ 40 = u \cdot 5 + \frac{1}{2} a \cdot (5^2) \]
This simplifies to:
\[ 40 = 5u + \frac{25}{2} a \]
Multiplying through by 2 to eliminate the fraction:
\[ 80 = 10u + 25a \quad \text{(Equation 1)} \]

### Step 3: Write the Equation for the Next 5 Seconds
The distance covered in the next 5 seconds is given as 65 m. The total distance covered in 10 seconds is:
\[ s_{total} = s_1 + s_2 = 40 + 65 = 105 \, \text{m} \]
Using the equation of motion for the entire 10 seconds:
\[ s_{total} = ut + \frac{1}{2} a t^2 \]
Substituting the known values:
\[ 105 = u \cdot 10 + \frac{1}{2} a \cdot (10^2) \]
This simplifies to:
\[ 105 = 10u + 50a \]
Multiplying through by 2:
\[ 210 = 20u + 100a \quad \text{(Equation 2)} \]

### Step 4: Solve the Equations Simultaneously
Now we have two equations:
1. \( 80 = 10u + 25a \)
2. \( 210 = 20u + 100a \)

We can simplify Equation 1 by multiplying it by 2:
\[ 160 = 20u + 50a \quad \text{(Equation 3)} \]

Now we can subtract Equation 3 from Equation 2:
\[ (210 - 160) = (20u + 100a) - (20u + 50a) \]
This simplifies to:
\[ 50 = 50a \]
So, we find:
\[ a = 1 \, \text{m/s}^2 \]

### Step 5: Substitute Back to Find Initial Velocity
Now substitute \( a \) back into Equation 1 to find \( u \):
\[ 80 = 10u + 25(1) \]
This simplifies to:
\[ 80 = 10u + 25 \]
Subtracting 25 from both sides:
\[ 55 = 10u \]
Dividing by 10:
\[ u = 5.5 \, \text{m/s} \]

### Final Answer
The initial velocity \( u \) is \( 5.5 \, \text{m/s} \).

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