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 find the relationship between the initial velocity, acceleration, and time for a particle whose acceleration is given by \( f = at \).

### Step 1: Understand the given information
- Initial velocity of the particle at \( t = 0 \) is \( u \).
- Acceleration is given by \( a(t) = at \), where \( a \) is a constant.

### Step 2: Write down the expression for acceleration
The acceleration of the particle is given by:
\[ a(t) = at \]

### Step 3: Relate acceleration to velocity
To find the velocity as a function of time, we need to integrate the acceleration with respect to time. The relationship between acceleration and velocity is given by:
\[ a = \frac{dv}{dt} \]
Substituting the expression for acceleration:
\[ \frac{dv}{dt} = at \]

### Step 4: Integrate the acceleration
We will integrate both sides with respect to time from \( t = 0 \) to \( t = t \):
\[ \int_{u}^{v} dv = \int_{0}^{t} at \, dt \]
The left side becomes:
\[ v - u \]
The right side can be integrated as follows:
\[ \int_{0}^{t} at \, dt = a \int_{0}^{t} t \, dt = a \left[ \frac{t^2}{2} \right]_{0}^{t} = a \frac{t^2}{2} \]

### Step 5: Combine the results
Now, equating both sides gives us:
\[ v - u = a \frac{t^2}{2} \]
Rearranging this equation, we find:
\[ v = u + a \frac{t^2}{2} \]

### Conclusion
The valid relation that describes the velocity of the particle at time \( t \) is:
\[ v = u + a \frac{t^2}{2} \]