Chapter 3, Problem 3.43P

Review. As it passes over Grand Bahama Island, the eye of a hurricane is moving in a direction 60.0° north of west with a speed of 41.0 km/h. (a) What is the unit-vector expression for the velocity of the hurricane? It maintains this velocity for 3.00 h. at which time the course of the hurricane suddenly shifts due north, and its speed slows to a constant 25.0 km/h. This new velocity is maintained for 1.50 h. (b) What is the unit-vector expression for the new velocity of the hurricane? (c) What is the unit-vector expression for the displacement of the hurricane during the first 3.00 h? (d) What is the unit-vector expression for the displacement of the hurricane during the latter 1.50 h? (e) How far from Grand Bahama is the eye 4.50 h after it passes over the island?

To determine

The unit vector expression for the velocity of hurricane during first $3$ hours.

Answer to Problem 3.43P

The unit vector expression for the velocity of hurricane during first $3$ hours is $\left(-20.5\text{km}/\text{h}\right)\widehat{i}+\left(35.50\text{km}/\text{h}\right)\widehat{j}$.

Explanation of Solution

Given information:

The velocity of hurricane for the first $3$ hours is $41.0\text{km}/\text{h}$ with a direction of $60.0°$ north of west and the velocity of hurricane for next $1.50$ hour is $25.0\text{km}/\text{h}$ with a direction along north.

Consider the east direction as positive $x$ direction.

The unit vector expression of velocity of hurricane for first $3$ hours is given as,

${\overrightarrow{\text{v}}}_1=\left(-\left|{\overrightarrow{\text{v}}}_{\text{1}}\right|\cos 60.0°\right)\widehat{i}+\left(\left|{\overrightarrow{\text{v}}}_{\text{1}}\right|\sin 60.0°\right)\widehat{j}$ (I)

Substitute $41.0\text{km}/\text{h}$ for $\left|{\overrightarrow{\text{v}}}_{\text{1}}\right|$ in above expression in equation (I).

$\begin{array}{c}{\overrightarrow{\text{v}}}_1=\left(-\left(41.0\text{km}/\text{h}\right)\cos 60.0°\right)\widehat{i}+\left(\left(41.0\text{km}/\text{h}\right)\sin 60.0°\right)\widehat{j}\\ =\left(-20.5\text{km}/\text{h}\right)\widehat{i}+\left(35.50\text{km}/\text{h}\right)\widehat{j}\end{array}$

Conclusion:

Therefore, the unit vector expression for the velocity of hurricane during first $3$ hours is $\left(-20.5\text{km}/\text{h}\right)\widehat{i}+\left(35.50\text{km}/\text{h}\right)\widehat{j}$.

To determine

The unit vector expression for the new velocity of the hurricane.

Answer to Problem 3.43P

The unit vector expression for the new velocity of hurricane is $\left(25.0\text{km}/\text{h}\right)\widehat{j}$.

Explanation of Solution

Given information:

The velocity of hurricane for the first $3$ hours is $41.0\text{km}/\text{h}$ with a direction of $60.0°$ north of west and the velocity of hurricane for next $1.50$ hour is $25.0\text{km}/\text{h}$ with a direction along north.

The unit vector expression for the new velocity of the hurricane is,

${\overrightarrow{\text{v}}}_{\text{2}}=\left(\left|{\overrightarrow{\text{v}}}_{\text{2}}\right|\right)\widehat{j}$

Substitute $25.0\text{km}/\text{h}$ for $\left|{\overrightarrow{\text{v}}}_{\text{2}}\right|$ in above expression.

${\overrightarrow{\text{v}}}_{\text{2}}=\left(25.0\text{km}/\text{h}\right)\widehat{j}$

Conclusion:

Therefore, the unit vector expression for the new velocity of hurricane is $\left(25.0\text{km}/\text{h}\right)\widehat{j}$.

To determine

The unit vector expression for the displacement of the hurricane during first $3$ hours.

Answer to Problem 3.43P

The unit vector expression for the displacement of the hurricane during first $3$ hours is $\left(-61.5\text{km}\right)\widehat{i}+\left(106.5\text{km}\right)\widehat{j}$.

Explanation of Solution

Given information:

The velocity of hurricane for the first $3$ hours is $41.0\text{km}/\text{h}$ with a direction of $60.0°$ north of west and the velocity of hurricane for next $1.50$ hour is $25.0\text{km}/\text{h}$ with a direction along north.

Formula to determine the unit vector expression for the displacement during first $3$ hours is,

${\overrightarrow{\text{d}}}_1=\left({\overrightarrow{\text{v}}}_1\right)\left(t\right)$ (II)

• ${\overrightarrow{\text{v}}}_1$ is the velocity of hurricane during first $3$ hours.
• $t$ is the time.
• ${\overrightarrow{\text{d}}}_1$ is the displacement of hurricane for $3$ hours.

Substitute $\left(-20.5\text{km}/\text{h}\right)\widehat{i}+\left(35.50\text{km}/\text{h}\right)\widehat{j}$ for ${\overrightarrow{\text{v}}}_1$ and $3.00\text{hours}$ for $t$ in equation (II).

$\begin{array}{c}{\overrightarrow{\text{d}}}_1=\left(\left(-20.5\text{km}/\text{h}\right)\widehat{i}+\left(35.50\text{km}/\text{h}\right)\widehat{j}\right)\left(3.00\text{hours}\right)\\ =\left(-61.5\text{km}\right)\widehat{i}+\left(106.5\text{km}\right)\widehat{j}\end{array}$

Conclusion:

Therefore, the unit vector expression for the displacement of the hurricane during first $3$ hours is $\left(-61.5\text{km}\right)\widehat{i}+\left(106.5\text{km}\right)\widehat{j}$.

To determine

The unit vector expression for the displacement of the hurricane during the later $1.50$ hours.

Answer to Problem 3.43P

The unit vector expression for the displacement of the hurricane during the later $1.50$ hours is $\left(37.5\text{km}\right)\widehat{j}$.

Explanation of Solution

Given information:

The velocity of hurricane for the first $3$ hours is $41.0\text{km}/\text{h}$ with a direction of $60.0°$ north of west and the velocity of hurricane for next $1.50$ hour is $25.0\text{km}/\text{h}$ with a direction along north.

Formula to determine the unit vector expression for the displacement during the later $1.50$ hours is,

${\overrightarrow{\text{d}}}_2=\left({\overrightarrow{\text{v}}}_2\right)\left(t\right)$ (III)

• ${\overrightarrow{\text{v}}}_1$ is the velocity of hurricane during the later $1.50$ hours.
• $t$ is the time.
• ${\overrightarrow{\text{d}}}_1$ is the displacement of hurricane for the later $1.50$ hours.

Substitute $\left(25.0\text{km}/\text{h}\right)\widehat{j}$ for ${\overrightarrow{\text{v}}}_2$ and $1.50\text{hours}$ for $t$ in equation (III).

$\begin{array}{c}{\overrightarrow{\text{d}}}_1=\left(\left(25.0\text{km}/\text{h}\right)\widehat{j}\right)\left(1.50\text{hours}\right)\\ =\left(37.5\text{km}\right)\widehat{j}\end{array}$

Conclusion:

Therefore, the unit vector expression for the displacement of the hurricane during the later $1.50$ hours is $\left(37.5\text{km}\right)\widehat{j}$.

To determine

The net displacement of the hurricane after $4.50\text{hours}$.

Answer to Problem 3.43P

The net displacement of the hurricane after $4.50\text{hours}$ is $157\text{km}$.

Explanation of Solution

Section 1:

To determine: The unit vector expression for the net displacement after $4.50\text{hours}$.

Answer: The unit vector expression for the net displacement after $4.50\text{hours}$ is $\left(-61.5\text{km}\right)\widehat{i}+\left(144\text{km}\right)\widehat{j}$.

Explanation:

Given information:

The velocity of hurricane for the first $3$ hours is $41.0\text{km}/\text{h}$ with a direction of $60.0°$ north of west and the velocity of hurricane for next $1.50$ hour is $25.0\text{km}/\text{h}$ with a direction along north.

Formula to determine the unit vector expression for the net displacement after $4.50\text{hours}$ is,

$\overrightarrow{\text{d}}={\overrightarrow{\text{d}}}_1+{\overrightarrow{\text{d}}}_2$ (IV)

Substitute $\left(-61.5\text{km}\right)\widehat{i}+\left(106.5\text{km}\right)\widehat{j}$ for ${\overrightarrow{\text{d}}}_1$ and $\left(37.5\text{km}\right)\widehat{j}$ for ${\overrightarrow{\text{d}}}_2$ in equation (IV).

$\begin{array}{c}\overrightarrow{\text{d}}=\left(-61.5\text{km}\right)\widehat{i}+\left(106.5\text{km}\right)\widehat{j}+\left(37.5\text{km}\right)\widehat{j}\\ =\left(-61.5\text{km}\right)\widehat{i}+\left(144\text{km}\right)\widehat{j}\end{array}$

Section 2:

To determine: The magnitude of the net displacement after $4.50\text{hours}$.

Answer: The magnitude of the net displacement after $4.50\text{hours}$ is $157\text{km}$.

Explanation:

Given information:

The velocity of hurricane for the first $3$ hours is $41.0\text{km}/\text{h}$ with a direction of $60.0°$ north of west and the velocity of hurricane for next $1.50$ hour is $25.0\text{km}/\text{h}$ with a direction along north.

From part (d), the unit vector expression for the net displacement after $4.50\text{hours}$ is given as,

$\overrightarrow{\text{d}}=\left(-61.5\text{km}\right)\widehat{i}+\left(144\text{km}\right)\widehat{j}$

Formula to calculate the magnitude of the net displacement after $4.50\text{hours}$ is,

$\left|\overrightarrow{\text{d}}\right|=\sqrt{{\left(d_{\text{x}}\right)}^2+{\left(d_{\text{y}}\right)}^2}$ (V)

Substitute $-61.5\text{km}$ for $d_{\text{x}}$ and $144\text{km}$ for $d_{\text{y}}$ in equation (V).

$\begin{array}{c}\left|\overrightarrow{\text{d}}\right|=\sqrt{{\left(-61.5\text{km}\right)}^2+{\left(144\text{km}\right)}^2}\\ =156.58\text{km}\\ \approx 157\text{km}\end{array}$

Conclusion:

Therefore, the magnitude of the net displacement after $4.50\text{hours}$ is $157\text{km}$