To solve the problem of how far the bowling ball traveled over a time duration of 1.40 seconds while speeding up from an initial velocity of 4.00 m/s to a final velocity of 7.20 m/s, we can follow these steps:
### Step 1: Identify Given Values
- Initial Velocity, [tex]\( v_i \)[/tex]: 4.00 m/s
- Final Velocity, [tex]\( v_f \)[/tex]: 7.20 m/s
- Time Duration, [tex]\( t \)[/tex]: 1.40 seconds
### Step 2: Calculate the Average Velocity
Since the velocity changes uniformly, the average velocity can be calculated using the formula:
[tex]\[ v_{\text{avg}} = \frac{v_i + v_f}{2} \][/tex]
Substitute the given values:
[tex]\[ v_{\text{avg}} = \frac{4.00 \text{ m/s} + 7.20 \text{ m/s}}{2} \][/tex]
[tex]\[ v_{\text{avg}} = \frac{11.20 \text{ m/s}}{2} \][/tex]
[tex]\[ v_{\text{avg}} = 5.60 \text{ m/s} \][/tex]
### Step 3: Calculate the Distance Traveled
Distance traveled can be found by multiplying the average velocity by the time duration:
[tex]\[ d = v_{\text{avg}} \times t \][/tex]
Substitute the calculated average velocity and the given time duration:
[tex]\[ d = 5.60 \text{ m/s} \times 1.40 \text{ s} \][/tex]
[tex]\[ d = 7.84 \text{ m} \][/tex]
Therefore, the bowling ball traveled a distance of 7.84 meters in 1.40 seconds.
