A body covers 20 m, 22 m, 24 m, in `8^(th)`, `9 ^(th)` and `10^(th)` seconds respectively. The body starts

Options

A. 1. from rest and moves with uniform velocity.
B. 2. from rest and moves with uniform acceleration.
C. 3. with an initial velocity and moves with uniform acceleration.
D. 4. with an initial velocity and moves with uniform velocity.

Explanation

To solve the problem step by step, we will use the formula for the distance covered by a body in the nth second of motion under uniform acceleration. The formula is:

\[ S_n = u + \frac{a}{2} (2n - 1) \]

where:
- \( S_n \) is the distance covered in the nth second,
- \( u \) is the initial velocity,
- \( a \) is the acceleration,
- \( n \) is the second in which the distance is covered.

### Step 1: Set up the equations for the distances covered in the 8th, 9th, and 10th seconds.

Given:
- \( S_8 = 20 \, m \)
- \( S_9 = 22 \, m \)
- \( S_{10} = 24 \, m \)

Using the formula for \( S_8 \):
\[ S_8 = u + \frac{a}{2} (2 \cdot 8 - 1) = u + \frac{a}{2} (16 - 1) = u + \frac{15a}{2} \]
Setting this equal to 20:
\[ u + \frac{15a}{2} = 20 \quad \text{(Equation 1)} \]

Using the formula for \( S_9 \):
\[ S_9 = u + \frac{a}{2} (2 \cdot 9 - 1) = u + \frac{a}{2} (18 - 1) = u + \frac{17a}{2} \]
Setting this equal to 22:
\[ u + \frac{17a}{2} = 22 \quad \text{(Equation 2)} \]

### Step 2: Subtract Equation 1 from Equation 2 to find acceleration \( a \).

Subtracting Equation 1 from Equation 2:
\[ \left(u + \frac{17a}{2}\right) - \left(u + \frac{15a}{2}\right) = 22 - 20 \]
This simplifies to:
\[ \frac{2a}{2} = 2 \implies a = 2 \, m/s^2 \]

### Step 3: Substitute the value of \( a \) back into Equation 1 to find \( u \).

Substituting \( a = 2 \) into Equation 1:
\[ u + \frac{15 \cdot 2}{2} = 20 \]
This simplifies to:
\[ u + 15 = 20 \implies u = 20 - 15 = 5 \, m/s \]

### Step 4: Verify the value of \( S_{10} \).

Now, we can verify if our values of \( u \) and \( a \) give us the correct distance for \( S_{10} \):
Using the formula for \( S_{10} \):
\[ S_{10} = u + \frac{a}{2} (2 \cdot 10 - 1) = 5 + \frac{2}{2} (20 - 1) = 5 + 19 = 24 \, m \]
This confirms that our calculations are correct.

### Conclusion

The body has an initial velocity of \( 5 \, m/s \) and an acceleration of \( 2 \, m/s^2 \). Therefore, the correct option is that the body is moving with an initial velocity and uniform acceleration.

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