Q: A hiker walks 36m north and then 28m east from the origin. What is the magnitude and direction of the hiker's displacement from the origin?

$$\sqrt{36^2$ + 28^2$} = 45.0\,\text{m}$$, $$\theta = \tan$$^{-1}$$\left(\frac{36}{28}\right) = 52.4^\circ$$ north of east

Q: Your initial position is 6.2m from the origin, 25° below the x-axis. You then travel 9.9m at an angle 78° above the positive x-axis, then 2.0m in the negative x-direction. What is the magnitude and direction of your final position vector?

$$10.4\,\text{m}$$, $$42.6^\circ$$ above the x-axis

Q: A person walks 40m in the +x direction, then 30m in the +y direction. The entire trip takes 10 seconds. What is the magnitude and direction of their average velocity?

$$\frac{\sqrt{40^2$ + 30^2$}}{10} = 5.0\,\text{m/s}$$, $$\theta = \tan$$^{-1}$$\left(\frac{30}{40}\right) = 36.9^\circ$$ above x-axis

Q: You walk 85m at 30.0° west of north, then walk 92m at 67.0° north of east. You reach the treasure 2 minutes later. What is your average speed for the entire trip?

$$\frac{85 + 92}{120} = 1.48\,\text{m/s}$$

Q: A ball moves on a tabletop. The ball has initial x & y coordinates (1.8m, 3.6m). The ball moves 10m/s at 53.1° above the x-axis for 4s. What are the x & y coordinates of the ball's final position?

$$(1.8 + 10 \cos 53.1^\circ \times 4,\ 3.6 + 10 \sin 53.1^\circ \times 4) = (33.8\,\text{m}, 27.6\,\text{m})$$

Q: A ball is launched at an angle of 40° above the ground (x-axis) with a velocity of 17m/s. What are the horizontal and vertical components of the launch velocity?

$$v_x = 17 \cos 40^\circ = 13.0\,\text{m/s}$$, $$v_y = 17 \sin 40^\circ = 10.9\,\text{m/s}$$

Q: A hockey puck slides along a lake at 8m/s east. A strong wind accelerates the puck at a constant 3m/s² in a direction 37° northeast. What is the magnitude and direction of the puck's displacement after 5s?

$$\Delta x = 8 \times 5 + \frac{1}{2} \times 3 \cos 37^\circ \times 5^2$, $$\Delta y = \frac{1}{2} \times 3 \sin 37^\circ \times $$5^2$$; $$|\Delta \vec{r}| = \sqrt{\Delta x^2$ + \Delta y^2$}$$, $$\theta = \tan$$^{-1}$$\left(\frac{\Delta y}{\Delta x}\right)$$

Q: A drone is located at (115, 72)m. If its initial velocity is 16m/s in the +y direction, and it has only 18s of battery life remaining, what constant acceleration (magnitude and direction) does it need to reach the lab at the origin?

$$a = \frac{2(0 - 72 - 16 \times 18)}{18^2$}$$, $$\theta = \tan$$^{-1}$$\left(\frac{0 - 115}{0 - 72}\right)$$

Q: You are an observer measuring the velocities of A, B, and C on a moving walkway moving at 3m/s. A stands still, B walks right at 2m/s, C walks left at 2m/s. What is the velocity of B relative to the ground?

$$v_{B,\text{ground}} = v_{B,\text{walkway}} + v_{\text{walkway,ground}} = 2 + 3 = 5\,\text{m/s}$$

Q: A car is moving at 45m/s east, relative to the ground. A truck ahead is moving at 60m/s east, also relative to the ground. What is the velocity of the truck relative to the car?

$$v_{\text{truck,car}} = v_{\text{truck,ground}} - v_{\text{car,ground}} = 60 - 45 = 15\,\text{m/s}$$ east

Question 1

A hiker walks 36m north and then 28m east from the origin. What is the magnitude and direction of the hiker's displacement from the origin?

Accepted Answer

$$\sqrt{36^2$ + 28^2$} = 45.0\,	ext{m}$$, $$	heta = 	an$$^{-1}$$\left(\frac{36}{28}ight) = 52.4^\circ$$ north of east

Question 2

Your initial position is 6.2m from the origin, 25° below the x-axis. You then travel 9.9m at an angle 78° above the positive x-axis, then 2.0m in the negative x-direction. What is the magnitude and direction of your final position vector?

Accepted Answer

$$10.4\,	ext{m}$$, $$42.6^\circ$$ above the x-axis

Question 3

A person walks 40m in the +x direction, then 30m in the +y direction. The entire trip takes 10 seconds. What is the magnitude and direction of their average velocity?

Accepted Answer

$$\frac{\sqrt{40^2$ + 30^2$}}{10} = 5.0\,	ext{m/s}$$, $$	heta = 	an$$^{-1}$$\left(\frac{30}{40}ight) = 36.9^\circ$$ above x-axis

Question 4

You walk 85m at 30.0° west of north, then walk 92m at 67.0° north of east. You reach the treasure 2 minutes later. What is your average speed for the entire trip?

Accepted Answer

$$\frac{85 + 92}{120} = 1.48\,	ext{m/s}$$

Question 5

A ball moves on a tabletop. The ball has initial x & y coordinates (1.8m, 3.6m). The ball moves 10m/s at 53.1° above the x-axis for 4s. What are the x & y coordinates of the ball's final position?

Accepted Answer

$$(1.8 + 10 \cos 53.1^\circ 	imes 4,\ 3.6 + 10 \sin 53.1^\circ 	imes 4) = (33.8\,	ext{m}, 27.6\,	ext{m})$$

Question 6

A ball is launched at an angle of 40° above the ground (x-axis) with a velocity of 17m/s. What are the horizontal and vertical components of the launch velocity?

Accepted Answer

$$v_x = 17 \cos 40^\circ = 13.0\,	ext{m/s}$$, $$v_y = 17 \sin 40^\circ = 10.9\,	ext{m/s}$$

Question 7

A hockey puck slides along a lake at 8m/s east. A strong wind accelerates the puck at a constant 3m/s² in a direction 37° northeast. What is the magnitude and direction of the puck's displacement after 5s?

Accepted Answer

$$\Delta x = 8 	imes 5 + \frac{1}{2} 	imes 3 \cos 37^\circ 	imes 5^2$, $$\Delta y = \frac{1}{2} 	imes 3 \sin 37^\circ 	imes $$5^2$$; $$|\Delta \vec{r}| = \sqrt{\Delta x^2$ + \Delta y^2$}$$, $$	heta = 	an$$^{-1}$$\left(\frac{\Delta y}{\Delta x}ight)$$

Question 8

A drone is located at (115, 72)m. If its initial velocity is 16m/s in the +y direction, and it has only 18s of battery life remaining, what constant acceleration (magnitude and direction) does it need to reach the lab at the origin?

Accepted Answer

$$a = \frac{2(0 - 72 - 16 	imes 18)}{18^2$}$$, $$	heta = 	an$$^{-1}$$\left(\frac{0 - 115}{0 - 72}ight)$$

Question 9

You are an observer measuring the velocities of A, B, and C on a moving walkway moving at 3m/s. A stands still, B walks right at 2m/s, C walks left at 2m/s. What is the velocity of B relative to the ground?

Accepted Answer

$$v_{B,	ext{ground}} = v_{B,	ext{walkway}} + v_{	ext{walkway,ground}} = 2 + 3 = 5\,	ext{m/s}$$

Question 10

A car is moving at 45m/s east, relative to the ground. A truck ahead is moving at 60m/s east, also relative to the ground. What is the velocity of the truck relative to the car?

Accepted Answer

$$v_{	ext{truck,car}} = v_{	ext{truck,ground}} - v_{	ext{car,ground}} = 60 - 45 = 15\,	ext{m/s}$$ east

Question 11

A boat travels downstream in a river that flows at 4m/s. If the boat makes the trip in 250s, what is the speed of the boat relative to the water?

Accepted Answer

$$v_{	ext{boat,water}} = v_{	ext{boat,ground}} - v_{	ext{water,ground}}$$

Question 12

If a steady 225-km/h wind blows from west to east and the airliner has the same speed relative to the air as before, how long will the trip from A to B take if the airliner travels 900km east?

Accepted Answer

$$t = \frac{900}{v_{	ext{plane,air}} + v_{	ext{wind,ground}}}$$

Question 13

Which of the following statements about vectors is correct?

Accepted Answer

A vector has both magnitude and direction, while a scalar has only magnitude.

Question 14

Which equation correctly gives the magnitude of a 2D displacement vector $$\Delta \vec{r}$$ with components $$\Delta x$$ and $$\Delta y$$?

Accepted Answer

$$|\Delta \vec{r}| = \sqrt{(\Delta x)^2$ + (\Delta y)^2$}$$

Question 15

A car accelerates from rest at 5m/s² for 4s in the direction 60° above the x-axis. What are the x and y components of its final velocity?

Accepted Answer

$$v_x = 5 	imes 4 \cos 60^\circ = 10\,	ext{m/s}$$, $$v_y = 5 	imes 4 \sin 60^\circ = 17.3\,	ext{m/s}$$

Question 16

Which of the following best describes the difference between speed and velocity?

Accepted Answer

Speed is a scalar quantity (magnitude only), while velocity is a vector quantity (magnitude and direction).

Question 17

A ball is pushed 650m west and 800m south in 600s. What is the magnitude and direction of your average velocity?

Accepted Answer

$$|\vec{v}_{	ext{avg}}| = \frac{\sqrt{650^2$ + 800^2$}}{600} = 2.15\,	ext{m/s}$$, $$	heta = 	an$$^{-1}$$\left(\frac{800}{650}ight) = 50.9^\circ$$ south of west

Question 18

A toy car is initially moving 20m/s in the +x-axis. 10 seconds later, the car is moving 67m/s at 26.5° above the x-axis. What are the x & y components of the car's acceleration over the 10s?

Accepted Answer

$$a_x = \frac{67 \cos 26.5^\circ - 20}{10}$$, $$a_y = \frac{67 \sin 26.5^\circ - 0}{10}$$

2D Kinematics and Relative Motion: Study Notes for College Physics

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