A body moving with uniform acceleration describes 12 m in the third second of its motion and 20 m in the fifth second. The velocity of the body after `10^(th)` s is 

Options

A. `40 " m s"^(-1)`
B. `42 " m s"^(-1)`
C. `52 " m s"^(-1)`
D. `4 " m s"^(-1)`

Explanation

To solve the problem step by step, we will use the equations of motion for a body moving with uniform acceleration. We are given that a body describes 12 m in the third second and 20 m in the fifth second of its motion. We want to find the velocity of the body after 10 seconds.

### Step 1: Understand the formula for displacement in the nth second
The displacement \( S_n \) in the nth second is given by the formula:
\[ S_n = u + \frac{a}{2} \cdot (2n - 1) \]
where:
- \( S_n \) is the displacement in the nth second,
- \( u \) is the initial velocity,
- \( a \) is the acceleration,
- \( n \) is the second.

### Step 2: Set up the equations for the given displacements
For the third second (\( n = 3 \)):
\[ S_3 = u + \frac{a}{2} \cdot (2 \cdot 3 - 1) = u + \frac{a}{2} \cdot 5 = 12 \quad \text{(1)} \]

For the fifth second (\( n = 5 \)):
\[ S_5 = u + \frac{a}{2} \cdot (2 \cdot 5 - 1) = u + \frac{a}{2} \cdot 9 = 20 \quad \text{(2)} \]

### Step 3: Solve the equations simultaneously
Now we have two equations:
1. \( u + \frac{5a}{2} = 12 \)
2. \( u + \frac{9a}{2} = 20 \)

We can subtract equation (1) from equation (2):
\[ \left(u + \frac{9a}{2}\right) - \left(u + \frac{5a}{2}\right) = 20 - 12 \]
This simplifies to:
\[ \frac{9a}{2} - \frac{5a}{2} = 8 \]
\[ \frac{4a}{2} = 8 \]
\[ 2a = 8 \quad \Rightarrow \quad a = 4 \, \text{m/s}^2 \]

### Step 4: Substitute the value of \( a \) back to find \( u \)
Now, substitute \( a = 4 \) back into equation (1):
\[ u + \frac{5 \cdot 4}{2} = 12 \]
\[ u + 10 = 12 \quad \Rightarrow \quad u = 12 - 10 = 2 \, \text{m/s} \]

### Step 5: Find the velocity after 10 seconds
Now we can find the velocity \( v \) after 10 seconds using the formula:
\[ v = u + at \]
Substituting the values we found:
\[ v = 2 + 4 \cdot 10 \]
\[ v = 2 + 40 = 42 \, \text{m/s} \]

### Final Answer
The velocity of the body after 10 seconds is \( 42 \, \text{m/s} \).

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