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

Answer

To solve the problem, we need to derive the relationship between the final velocity \( v \) of a particle, its initial velocity \( u \), and the acceleration \( a \) which is given as \( f = at \).

### Step-by-Step Solution:

1. **Understand the Given Information**:
- Initial velocity at \( t = 0 \) is \( u \).
- Acceleration is given by \( f = at \).

2. **Relate Acceleration to Velocity**:
- We know that acceleration \( a \) is defined as the rate of change of velocity with respect to time:
\[ a = \frac{dv}{dt} \]
- Rearranging gives:
\[ dv = a \, dt \]

3. **Substitute the Expression for Acceleration**:
- Since \( f = at \), we can substitute \( a \) in the equation:
\[ dv = (f) \, dt = (at) \, dt \]

4. **Integrate Both Sides**:
- Integrate the left side with respect to \( v \) and the right side with respect to \( t \):
\[ \int dv = \int at \, dt \]
- The left side integrates to \( v \), and the right side integrates to \( \frac{1}{2}at^2 \):
\[ v = \frac{1}{2}at^2 + C \]
- Here, \( C \) is the constant of integration.

5. **Determine the Constant of Integration**:
- At \( t = 0 \), the velocity \( v = u \):
\[ u = \frac{1}{2}a(0)^2 + C \implies C = u \]

6. **Final Expression for Velocity**:
- Substitute \( C \) back into the equation:
\[ v = u + \frac{1}{2}at^2 \]

7. **Conclusion**:
- The valid relation that describes the motion of the particle is:
\[ v = u + \frac{1}{2}at^2 \]