The initial velocity of a particle is u (at t = 0) and the acceleration f is given by f = at. Which of the following relation is valid?

Options

A. `v = u + at^(2)`
B. `v = u + (at^(2))/(2)`
C. `v = u + at`
D. v = u

Explanation

To solve the problem step by step, we need to analyze the given information and derive the relationship between the initial velocity, acceleration, and final velocity of the particle.

### Step-by-Step Solution:

1. **Identify the Given Information:**
- Initial velocity of the particle, \( u \) (at \( t = 0 \)).
- Acceleration, \( f = a t \).

2. **Understand the Relationship Between Acceleration and Velocity:**
- Acceleration is defined as the rate of change of velocity with respect to time:
\[ a = \frac{dv}{dt} \]
- Given that \( f = a t \), we can substitute \( a \) into the equation:
\[ \frac{dv}{dt} = a t \]

3. **Rearranging the Equation:**
- We can rearrange the equation to isolate \( dv \):
\[ dv = a t \, dt \]

4. **Integrate Both Sides:**
- We need to integrate both sides. The left side will be integrated from the initial velocity \( u \) to the final velocity \( v \), and the right side will be integrated from \( 0 \) to \( t \):
\[ \int_{u}^{v} dv = \int_{0}^{t} a t \, dt \]

5. **Perform the Integration:**
- The left side becomes:
\[ v - u \]
- The right side involves integrating \( a t \):
\[ \int_{0}^{t} a t \, dt = a \int_{0}^{t} t \, dt = a \left[ \frac{t^2}{2} \right]_{0}^{t} = a \frac{t^2}{2} \]

6. **Set the Integrated Results Equal:**
- Now we equate both sides:
\[ v - u = a \frac{t^2}{2} \]

7. **Solve for Final Velocity \( v \):**
- Rearranging the equation gives us:
\[ v = u + a \frac{t^2}{2} \]

### Final Result:
The valid relation is:
\[ v = u + a \frac{t^2}{2} \]