Question

1. a plane flies 100 km north, then 75 km east. find the resultant displacement (magnitude and direction).
2. a hiker walks 3 km west, then 4 km north. what is the resultant displacement (magnitude and angle)?
3. a boat heads 8 km south, then 6 km east. find the resultant displacement using the pythagorean theorem and trig ratios.
4. a person pushes a box with a force of 20 n east, while another person pushes with 15 n north. find the resultant force (magnitude and direction).
5. a rescue helicopter flies 150 m west, then 80 m north. calculate the helicopters displacement (magnitude and angle).

Explanation:

5.

Step1: Identify components

Let the north - south displacement be $y = 100$ km and the east - west displacement be $x=75$ km.

Step2: Calculate magnitude

Use the Pythagorean theorem $R=\sqrt{x^{2}+y^{2}}$. Substitute $x = 75$ and $y = 100$: $R=\sqrt{75^{2}+100^{2}}=\sqrt{5625 + 10000}=\sqrt{15625}=125$ km.

Step3: Calculate direction

Use $\theta=\tan^{- 1}(\frac{y}{x})$. Substitute $x = 75$ and $y = 100$, $\theta=\tan^{-1}(\frac{100}{75})=\tan^{-1}(\frac{4}{3})\approx53.13^{\circ}$ north of east.

Step1: Identify components

Let the west - east displacement be $x=- 3$ km and the north - south displacement be $y = 4$ km.

Step2: Calculate magnitude

Use the Pythagorean theorem $R=\sqrt{x^{2}+y^{2}}$. Substitute $x=-3$ and $y = 4$: $R=\sqrt{(-3)^{2}+4^{2}}=\sqrt{9 + 16}=\sqrt{25}=5$ km.

Step3: Calculate angle

Use $\theta=\tan^{-1}(\frac{y}{|x|})$. Substitute $x=-3$ and $y = 4$, $\theta=\tan^{-1}(\frac{4}{3})\approx53.13^{\circ}$ north of west.

Step1: Identify components

Let the south - north displacement be $y=-8$ km and the east - west displacement be $x = 6$ km.

Step2: Calculate magnitude

Use the Pythagorean theorem $R=\sqrt{x^{2}+y^{2}}$. Substitute $x = 6$ and $y=-8$: $R=\sqrt{6^{2}+(-8)^{2}}=\sqrt{36 + 64}=\sqrt{100}=10$ km.

Step3: Calculate angle

Use $\theta=\tan^{-1}(\frac{|y|}{x})$. Substitute $x = 6$ and $y=-8$, $\theta=\tan^{-1}(\frac{8}{6})=\tan^{-1}(\frac{4}{3})\approx53.13^{\circ}$ south of east.

Answer:

Magnitude: 125 km, Direction: $53.13^{\circ}$ north of east

6.