To determine the time taken for the boat to travel a distance of 48 meters as it speeds up from an initial speed of [tex]\(5.2 \, \text{m/s}\)[/tex] to a final speed of [tex]\(9.8 \, \text{m/s}\)[/tex], we can follow these steps:
1. Calculate the average speed:
Since the boat is uniformly accelerating, the average speed [tex]\(\bar{v}\)[/tex] can be found by averaging the initial speed [tex]\(v_i\)[/tex] and the final speed [tex]\(v_f\)[/tex]:
[tex]\[
\bar{v} = \frac{v_i + v_f}{2}
\][/tex]
Given:
[tex]\[
v_i = 5.2 \, \text{m/s}
\][/tex]
[tex]\[
v_f = 9.8 \, \text{m/s}
\][/tex]
Therefore:
[tex]\[
\bar{v} = \frac{5.2 \, \text{m/s} + 9.8 \, \text{m/s}}{2} = \frac{15}{2} = 7.5 \, \text{m/s}
\][/tex]
2. Calculate the time taken:
Using the formula for time, [tex]\(t\)[/tex], which is the distance [tex]\(d\)[/tex] divided by the average speed [tex]\(\bar{v}\)[/tex]:
[tex]\[
t = \frac{d}{\bar{v}}
\][/tex]
Given the distance:
[tex]\[
d = 48 \, \text{meters}
\][/tex]
And the average speed:
[tex]\[
\bar{v} = 7.5 \, \text{m/s}
\][/tex]
Therefore:
[tex]\[
t = \frac{48 \, \text{meters}}{7.5 \, \text{m/s}} = 6.4 \, \text{seconds}
\][/tex]
So, the time it took for the boat to travel the 48 meters is [tex]\( t = 6.4 \)[/tex] seconds.
