The motion of a particle is described by the equation at `u = at`.The distance travelled by the particle in the first 4 seconds

Options

A. `4 a`
B. `12 a`
C. `6 a`
D. `8 a`

Explanation

To solve the problem, we need to find the distance traveled by a particle in the first 4 seconds, given that its motion is described by the equation \( u = At \).

### Step-by-Step Solution:

1. **Identify the Given Information:**
- The equation of motion is given as \( u = At \).
- We need to find the distance traveled in the first 4 seconds.

2. **Determine the Acceleration:**
- The velocity \( u \) is a function of time \( t \).
- To find the acceleration, we differentiate the velocity with respect to time:
\[ a = \frac{du}{dt} = \frac{d(At)}{dt} = A \]
- Thus, the acceleration \( a \) is constant and equal to \( A \).

3. **Use the Equation of Motion:**
- The equation for distance \( s \) traveled under constant acceleration is:
\[ s = ut + \frac{1}{2}at^2 \]
- Here, \( u \) is the initial velocity, \( a \) is the acceleration, and \( t \) is the time.

4. **Find the Initial Velocity:**
- To find the initial velocity \( u \) at \( t = 0 \):
\[ u = A \cdot 0 = 0 \]
- Therefore, the initial velocity \( u = 0 \).

5. **Substitute Values into the Equation:**
- Now substitute \( u = 0 \), \( a = A \), and \( t = 4 \) seconds into the equation:
\[ s = 0 \cdot 4 + \frac{1}{2}A(4^2) \]
- Simplifying this gives:
\[ s = \frac{1}{2}A \cdot 16 = 8A \]

6. **Conclusion:**
- The distance traveled by the particle in the first 4 seconds is:
\[ s = 8A \]

### Final Answer:
The distance traveled by the particle in the first 4 seconds is \( 8A \).

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