Ty Ridlegs boards a paddle boat and heads the boat north at 9 m/s, directly across a river.  The river flows east at 14 m/s. Determine the magnitude and direction of the Ty's resultant velocity.



To determine Ty Ridlegs' resultant velocity, we can treat this as a vector problem. Ty’s velocity northward and the river's velocity eastward form two perpendicular vectors, so we can use the Pythagorean theorem and trigonometry to find the magnitude and direction of the resultant velocity.

### Step 1: Given Data
- Ty’s northward velocity (boat speed) = \( 9 \, \text{m/s} \)
- River’s eastward velocity (current speed) = \( 14 \, \text{m/s} \)

### Step 2: Find the Magnitude of the Resultant Velocity
The magnitude of the resultant velocity is the hypotenuse of a right triangle, where the boat's velocity and the river's velocity are the two legs. Using the Pythagorean theorem:

\[
v_{\text{resultant}} = \sqrt{(v_{\text{north}})^2 + (v_{\text{east}})^2}
\]
\[
v_{\text{resultant}} = \sqrt{(9)^2 + (14)^2}
\]
\[
v_{\text{resultant}} = \sqrt{81 + 196}
\]
\[
v_{\text{resultant}} = \sqrt{277} \approx 16.64 \, \text{m/s}
\]

### Step 3: Find the Direction of the Resultant Velocity
The direction of the resultant velocity is the angle \( \theta \) that the velocity vector makes with the northward direction. We can find this angle using the tangent function:

\[
\tan \theta = \frac{v_{\text{east}}}{v_{\text{north}}}
\]
\[
\tan \theta = \frac{14}{9}
\]
\[
\theta = \tan^{-1} \left( \frac{14}{9} \right)
\]
\[
\theta \approx 57.53^\circ
\]

Thus, the resultant velocity is approximately \( 16.64 \, \text{m/s} \) at an angle of \( 57.53^\circ \) east of north.

### Final Answer:
- Magnitude of the resultant velocity: \( 16.64 \, \text{m/s} \)
- Direction of the resultant velocity: \( 57.53^\circ \) east of north

Would you like further details or explanations? Here are some related questions:

1. How does the magnitude of the resultant velocity change if Ty paddles at 12 m/s instead?
2. How would the result change if the river flows west instead of east?
3. What is the effect on direction if the river's speed increases?
4. How would Ty’s velocity be affected if the boat’s speed decreases to 5 m/s?
5. How can we apply this problem-solving approach to an airplane flying in a crosswind?

**Tip:** When solving vector problems, break the motion into perpendicular components, and use trigonometry to combine them.

Ty Ridlegs boards a paddle boat and heads the boat north at 9 m/s, directly across a river. The river flows east at 14 m/s. Determine the magnitude and direction of Ty's resultant velocity.

Learn how to calculate the resultant velocity of a boat traveling northward at 9 m/s across a river flowing eastward at 14 m/s. This solution involves vector addition, the Pythagorean Theorem, and trigonometry to find the magnitude and direction. Suitable for Grades 10-12 students.

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