A motorboat starting from rest on a lake accelerates in a straight line at a constant rate of `3.0 m//s^(2)` for `8.0 s`. How far does the boat travel during this time ?

Explanation

To solve the problem of how far the motorboat travels during its acceleration, we can use the kinematic equation for uniformly accelerated motion. The equation we will use is:

\[ s = ut + \frac{1}{2} a t^2 \]

Where:
- \( s \) = displacement (distance traveled)
- \( u \) = initial velocity
- \( a \) = acceleration
- \( t \) = time

### Step-by-Step Solution:

1. **Identify the given values:**
- Initial velocity \( u = 0 \) m/s (the boat starts from rest)
- Acceleration \( a = 3.0 \) m/s²
- Time \( t = 8.0 \) s

2. **Substitute the values into the equation:**
\[ s = ut + \frac{1}{2} a t^2 \]
Plugging in the values:
\[ s = (0)(8.0) + \frac{1}{2} (3.0) (8.0)^2 \]

3. **Calculate \( t^2 \):**
\[ (8.0)^2 = 64 \]

4. **Calculate \( \frac{1}{2} a t^2 \):**
\[ \frac{1}{2} (3.0) (64) = \frac{3.0 \times 64}{2} = \frac{192}{2} = 96 \]

5. **Combine the results:**
Since \( ut = 0 \), we have:
\[ s = 0 + 96 = 96 \text{ m} \]

6. **Final result:**
The motorboat travels a distance of \( 96 \) meters during the \( 8.0 \) seconds of acceleration.

### Summary:
The motorboat travels a distance of **96 meters** while accelerating at a constant rate of **3.0 m/s²** for **8.0 seconds**.