The intial velocity of a particle is u (at t=0) and the acceleration is given by f=at. Which of the following relations 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, we need to find the relationship between the velocity \( v \) and the given acceleration \( f = at \).

### Step-by-Step Solution:

1. **Given Data:**
- Initial velocity \( u \) at \( t = 0 \).
- Acceleration \( f = at \).

2. **Acceleration Definition:**
- Acceleration \( a \) is the rate of change of velocity with respect to time.
- Mathematically, \( a = \frac{dv}{dt} \).

3. **Substitute the Given Acceleration:**
- Given \( f = at \), so \( a = at \).

4. **Set Up the Differential Equation:**
- \( \frac{dv}{dt} = at \).

5. **Separate the Variables:**
- \( dv = at \, dt \).

6. **Integrate Both Sides:**
- Integrate with respect to \( t \) from 0 to \( t \) and with respect to \( v \) from \( u \) to \( v \):
\[ \int_{u}^{v} dv = \int_{0}^{t} at \, dt \]

7. **Solve the Integrals:**
- Left side: \( \int_{u}^{v} dv = v - u \).
- Right side: \( \int_{0}^{t} at \, dt = a \int_{0}^{t} t \, dt = a \left[ \frac{t^2}{2} \right]_{0}^{t} = a \left( \frac{t^2}{2} - \frac{0^2}{2} \right) = \frac{at^2}{2} \).

8. **Combine the Results:**
- \( v - u = \frac{at^2}{2} \).

9. **Solve for \( v \):**
- \( v = u + \frac{at^2}{2} \).

### Final Answer:
The valid relation is:
\[ v = u + \frac{at^2}{2} \]