Question

29. A boat is traveling east at a speed of 3.8 m/s. A person walks across the boat with a velocity of 1.3 m/s south. a. What is the person's speed relative to the water? b. In what direction, relative to the ground, does the person walk?​

Answer 1

a. To find the person's speed relative to the water, you can use the vector addition of their velocity.

The person is walking south with a velocity of 1.3 m/s, and the boat is traveling east at a velocity of 3.8 m/s. To find the person's speed relative to the water, we can treat these velocities as vectors in a right triangle, where the horizontal component is the boat's velocity (3.8 m/s) and the vertical component is the person's velocity (-1.3 m/s, considering south as the negative y-direction).

Using the Pythagorean theorem, we can find the magnitude of their combined velocity:

\[V_{\text{relative}} = \sqrt{(3.8 \, \text{m/s})^2 + (-1.3 \, \text{m/s})^2}\]

\[V_{\text{relative}} = \sqrt{14.44 + 1.69}\]

\[V_{\text{relative}} = \sqrt{16.13} \approx 4.01 \, \text{m/s}\]

So, the person's speed relative to the water is approximately 4.01 m/s.

b. To determine the direction in which the person walks relative to the ground, you can find the angle (θ) that their velocity vector makes with respect to the east direction (the boat's direction).

Use the inverse tangent (arctan) function to find this angle:

\[\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{1.3 \, \text{m/s}}{3.8 \, \text{m/s}}\]

\[\theta = \arctan\left(\frac{1.3}{3.8}\right)\]

Using a calculator, you can find the value of θ:

\[\theta \approx 19.2^\circ\]

So, relative to the ground, the person is walking at an angle of approximately 19.2 degrees south of east.

Answer 2

The person's speed relative to the water is 1.3 m/s south-east. The person walks in a direction of 1.3 m/s south-east relative to the ground.

Step-by-step explanation:

The person's speed relative to the water can be found by subtracting the velocity of the boat from the velocity of the person. In this case, the person's velocity relative to the water is 1.3 m/s south minus 3.8 m/s east, which results in a velocity of 1.3 m/s south-east.

The direction in which the person walks relative to the ground can be determined by considering the combined velocities of the boat and the person. The boat is moving east at 3.8 m/s and the person is moving south at 1.3 m/s. By adding these velocities, we can find that the person walks in a direction of 1.3 m/s south-east relative to the ground.