Certainly! Let's break down the problem step by step and find the resultant velocity of the boat as it moves across the river, considering the given velocities.
1. Understanding the Problem:
- The boat is traveling north with a speed of [tex]\( 2.40 \, \text{m/s} \)[/tex].
- The river is flowing east with a speed of [tex]\( 1.0 \, \text{m/s} \)[/tex].
2. Visualizing the Velocities:
Here, we have two perpendicular velocity vectors:
- One vector representing the boat's motion northward with a magnitude of [tex]\( 2.40 \, \text{m/s} \)[/tex].
- Another vector representing the river's flow eastward with a magnitude of [tex]\( 1.0 \, \text{m/s} \)[/tex].
3. Resultant Velocity:
Since these two velocities are perpendicular to each other, we can find the resultant velocity by constructing a right triangle where:
- The northward velocity ([tex]\( 2.40 \, \text{m/s} \)[/tex]) is one leg of the triangle.
- The eastward velocity ([tex]\( 1.0 \, \text{m/s} \)[/tex]) is the other leg of the triangle.
The hypotenuse of this right triangle will represent the resultant velocity.
4. Applying Pythagoras’ Theorem:
According to Pythagoras' theorem:
[tex]\[
\text{Resultant Velocity} = \sqrt{(\text{Northward Velocity})^2 + (\text{Eastward Velocity})^2}
\][/tex]
Substituting the given values:
[tex]\[
\text{Resultant Velocity} = \sqrt{(2.40 \, \text{m/s})^2 + (1.0 \, \text{m/s})^2}
\][/tex]
5. Calculating the Resultant Velocity:
[tex]\[
\text{Resultant Velocity} = \sqrt{5.76 + 1.0} = \sqrt{6.76} = 2.6 \, \text{m/s}
\][/tex]
6. Conclusion:
Therefore, the resultant velocity of the boat as it moves across the river is [tex]\( 2.6 \, \text{m/s} \)[/tex].
So, the final answer is:
The resultant velocity of the boat is [tex]\( 2.6 \, \text{m/s} \)[/tex].
