The initial velocity of a particle is u and the accelertion at time t later is given by at, then its velocity is given by-

Options

A. `u+(at^(2))/(2)`
B. `u^(2)+(at^(2))/(2)`
C. `(u+at^(2))/(2)`
D. none of these

Explanation

To solve the problem step by step, we start with the information given:

1. **Identify the Variables**:
- Initial velocity of the particle: \( u \)
- Acceleration at time \( t \): \( a(t) = At \) (where \( A \) is a constant)

2. **Write the Relationship Between Acceleration and Velocity**:
- Acceleration is defined as the rate of change of velocity with respect to time, which can be expressed mathematically as:
\[ a(t) = \frac{dv}{dt} \]
- Substituting the given acceleration into the equation, we have:
\[ \frac{dv}{dt} = At \]

3. **Integrate Both Sides**:
- To find the velocity \( v \), we need to integrate the equation with respect to time \( t \). We integrate both sides from \( t = 0 \) to \( t = t \) and from \( v = u \) to \( v = v \):
\[ \int_{u}^{v} dv = \int_{0}^{t} At \, dt \]

4. **Perform the Integration**:
- The left side integrates to:
\[ v - u \]
- The right side integrates as follows:
\[ \int_{0}^{t} At \, dt = A \int_{0}^{t} t \, dt = A \left[ \frac{t^2}{2} \right]_{0}^{t} = A \cdot \frac{t^2}{2} \]
- Therefore, we have:
\[ v - u = A \cdot \frac{t^2}{2} \]

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

6. **Final Expression for Velocity**:
- Thus, the final expression for the velocity of the particle at time \( t \) is:
\[ v = u + \frac{At^2}{2} \]