The distance x covered in time t by a body having initial velocity `v_(0)` and havig a constant acceleration a is given by `x=v_(0)t+((1)/(2))at^(2)`. This result follows from:-

Options

A. Newton's first law
B. Newton's second law
C. Newtons' third law
D. None of these three

Explanation

To solve the question regarding the distance \( x \) covered by a body with an initial velocity \( v_0 \) and constant acceleration \( a \), let's analyze the equation given:

\[ x = v_0 t + \frac{1}{2} a t^2 \]

This equation is derived from the basic principles of motion under constant acceleration. We will explore how this result follows from Newton's laws of motion, particularly focusing on the second law.

### Step-by-Step Solution:

1. **Understanding the Equation**:
The equation \( x = v_0 t + \frac{1}{2} a t^2 \) represents the distance covered by an object in time \( t \) when it starts with an initial velocity \( v_0 \) and accelerates at a constant rate \( a \).

2. **Deriving Velocity**:
To find the relationship between distance, velocity, and acceleration, we can differentiate the distance equation with respect to time \( t \):
\[ \frac{dx}{dt} = v_0 + at \]
Here, \( \frac{dx}{dt} \) gives us the velocity \( v \) of the object at any time \( t \).

3. **Deriving Acceleration**:
Next, we differentiate the velocity equation again to find acceleration:
\[ \frac{d^2x}{dt^2} = a \]
This shows that the acceleration \( a \) is constant, which is a key assumption in deriving the original equation.

4. **Relating to Newton's Second Law**:
According to Newton's second law, the net force \( F \) acting on an object is equal to the mass \( m \) of the object multiplied by its acceleration \( a \):
\[ F = ma \]
The acceleration \( a \) can be expressed as the change in velocity over time, which is consistent with our previous derivation of velocity and acceleration from the distance equation.

5. **Conclusion**:
The equation \( x = v_0 t + \frac{1}{2} a t^2 \) is derived from the principles of kinematics, which are fundamentally based on Newton's second law of motion. Therefore, the correct answer to the question is that this result follows from **Newton's second law**.

### Final Answer:
The result \( x = v_0 t + \frac{1}{2} a t^2 \) follows from **Newton's second law**.