Question

starting from one shore, you row a boat across a narrow river to the other shore. the river is 30.0 m wide. as you row, the river current moves your boat down the river a distance of 15.0 m. what is the resultant displacement of your boat? express your answer in components, and then determine the magnitude.

Explanation:

Step1: Define the components

Let the width of the river be the x - component ($x = 30.0$ m) and the distance moved downstream be the y - component ($y=15.0$ m).

Step2: Write the displacement vector in component form

The displacement vector $\vec{d}$ can be written as $\vec{d}=x\hat{i}+y\hat{j}$, so $\vec{d}=30.0\hat{i}+ 15.0\hat{j}$ m.

Step3: Calculate the magnitude

The magnitude of a two - dimensional vector $\vec{d}=x\hat{i}+y\hat{j}$ is given by $d = \sqrt{x^{2}+y^{2}}$. Substitute $x = 30.0$ m and $y = 15.0$ m into the formula:
\[

$$\begin{align*} d&=\sqrt{(30.0)^{2}+(15.0)^{2}}\\ &=\sqrt{900 + 225}\\ &=\sqrt{1125}\\ &\approx33.54\text{ m} \end{align*}$$

\]

Answer:

Component form: $\vec{d}=30.0\hat{i}+15.0\hat{j}$ m, Magnitude: $d\approx33.54$ m