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

Options

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

Answer

To solve the problem, we need to analyze the motion of a particle with an initial velocity \( u \) and an acceleration that varies with time, given by \( a = f(t) = at \). Here’s the step-by-step solution:

### Step 1: Understanding the Initial Conditions
The initial velocity of the particle at time \( t = 0 \) is given as \( u \).

### Step 2: Expressing Acceleration
The acceleration \( a \) is given as a function of time:
\[ a = f(t) = at \]
This means that the acceleration is directly proportional to time.

### Step 3: Finding the Velocity
To find the final velocity \( v \) at any time \( t \), we use the relation:
\[ v = u + \int_0^t a \, dt \]
Substituting the expression for acceleration:
\[ v = u + \int_0^t (at) \, dt \]

### Step 4: Performing the Integration
Now, we perform the integration:
\[ \int_0^t (at) \, dt = a \int_0^t t \, dt = a \left[ \frac{t^2}{2} \right]_0^t = a \cdot \frac{t^2}{2} \]
Thus, the integral evaluates to \( \frac{at^2}{2} \).

### Step 5: Final Expression for Velocity
Substituting back into the velocity equation:
\[ v = u + \frac{at^2}{2} \]

### Step 6: Conclusion
The relation that describes the final velocity \( v \) of the particle as a function of time \( t \) is:
\[ v = u + \frac{at^2}{2} \]
This is the valid relation for the given conditions.