A boat is traveling relative to the water at a speed of 5.0 m/s due south. Relative to the boat, a passenger walks toward the back of the boat at a speed of 1.5 m/s. What is the magnitude and direction of the passenger's velocity relative to the water?

Options

A. 5.2 m/s, south
B. 3.5 m/s, south
C. 3.5 m/s, north
D. 6.5 m/s, south

Answer

To solve the problem step by step, we need to find the velocity of the passenger relative to the water. Here’s how we can do it:

### Step 1: Define the velocities
1. **Velocity of the boat relative to water (Vb/w)**: The boat is traveling due south at a speed of 5.0 m/s. In vector notation, we can represent this as:
\[ \vec{V}_{b/w} = 0 \hat{i} - 5.0 \hat{j} \, \text{m/s} \]
(where \(\hat{j}\) is the unit vector in the north-south direction, with south being negative).

2. **Velocity of the passenger relative to the boat (Vp/b)**: The passenger is walking toward the back of the boat (which is also south) at a speed of 1.5 m/s. In vector notation, this can be represented as:
\[ \vec{V}_{p/b} = 0 \hat{i} - 1.5 \hat{j} \, \text{m/s} \]

### Step 2: Use the relative velocity equation
To find the velocity of the passenger relative to the water (Vp/w), we use the following equation:
\[ \vec{V}_{p/w} = \vec{V}_{p/b} + \vec{V}_{b/w} \]

### Step 3: Substitute the values
Substituting the values we have:
\[ \vec{V}_{p/w} = (0 \hat{i} - 1.5 \hat{j}) + (0 \hat{i} - 5.0 \hat{j}) \]
This simplifies to:
\[ \vec{V}_{p/w} = 0 \hat{i} - (1.5 + 5.0) \hat{j} = 0 \hat{i} - 6.5 \hat{j} \, \text{m/s} \]

### Step 4: Calculate the magnitude and direction
1. **Magnitude**: The magnitude of the velocity of the passenger relative to the water is:
\[ |\vec{V}_{p/w}| = 6.5 \, \text{m/s} \]

2. **Direction**: Since the velocity vector is negative in the j direction, it indicates that the direction is south.

### Final Answer
The magnitude of the passenger's velocity relative to the water is **6.5 m/s** and the direction is **south**.

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