Q: A hiker walks 3 km east and then 4 km north. What is the magnitude of the hiker’s net displacement?

$$\sqrt{3^2$ + 4^2$} = 5\ \text{km}$$

Q: A car drives up a slope at a constant speed of $$15\ \text{m/s}$$ for $$10\ \text{s}. $$The slope is at an angle $$\theta = 30^\circ. $$How much height does the car gain?

$$15 \times 10 \times \sin 30^\circ = 75\ \text{m}$$

Q: A dog runs off a dock at $$8.5\ \text{m/s}$$ horizontally. The dock is $$0.61\ \text{m}$$ above the water. How far does the dog travel horizontally before landing?

$$8.5 \times \sqrt{\frac{2 \times 0.61}{9.8}} \approx 2.1\ \text{m}$$

Q: For a projectile launched at speed $$v_0$$ and angle $$\theta$$ above the horizontal, what is the horizontal range $$R ($$neglecting air resistance)?

$$R = \frac{v_0$^2 \sin 2\theta}{g}$$

Q: A table saw blade has a diameter of $$0.25\ \text{m}$$ and spins at $$3600\ \text{rpm}. $$What is the speed of a tooth at the edge of the blade?

$$v = 2\pi r \times \frac{3600}{60} = 47.1\ \text{m/s}$$

Q: A skater moves in a circle of radius $$11\ \text{m} at a $$speed of $$v = 11.1\ \text{m/s}. $$What is the magnitude of her centripetal acceleration?

$$a_c = \frac{v^2$}{r} = \frac{(11.1)^2$}{11} = 11.2\ \text{m/s}^2$$

Question 1

Which of the following statements about vectors is correct?

Accepted Answer

A vector is a quantity that has both magnitude and direction.

Question 2

A hiker walks 3 km east and then 4 km north. What is the magnitude of the hiker’s net displacement?

Accepted Answer

$$\sqrt{3^2$ + 4^2$} = 5\ 	ext{km}$$

Question 3

If a vector $$\vec{A} is $$multiplied by a negative scalar $$-k$$, what happens to the vector?

Accepted Answer

Its direction is reversed and its magnitude is multiplied by $$k.$$

Question 4

A car drives up a slope at a constant speed of $$15\ 	ext{m/s}$$ for $$10\ 	ext{s}. $$The slope is at an angle $$	heta = 30^\circ. $$How much height does the car gain?

Accepted Answer

$$15 	imes 10 	imes \sin 30^\circ = 75\ 	ext{m}$$

Question 5

Which statement best describes projectile motion?

Accepted Answer

The horizontal and vertical components of motion are independent of each other.

Question 6

A dog runs off a dock at $$8.5\ 	ext{m/s}$$ horizontally. The dock is $$0.61\ 	ext{m}$$ above the water. How far does the dog travel horizontally before landing?

Accepted Answer

$$8.5 	imes \sqrt{\frac{2 	imes 0.61}{9.8}} \approx 2.1\ 	ext{m}$$

Question 7

For a projectile launched at speed $$v_0$$ and angle $$	heta$$ above the horizontal, what is the horizontal range $$R ($$neglecting air resistance)?

Accepted Answer

$$R = \frac{v_0$^2 \sin 2	heta}{g}$$

Question 8

A table saw blade has a diameter of $$0.25\ 	ext{m}$$ and spins at $$3600\ 	ext{rpm}. $$What is the speed of a tooth at the edge of the blade?

Accepted Answer

$$v = 2\pi r 	imes \frac{3600}{60} = 47.1\ 	ext{m/s}$$

Question 9

In uniform circular motion, what is the direction of the acceleration vector?

Accepted Answer

Toward the center of the circle

Question 10

A skater moves in a circle of radius $$11\ 	ext{m} at a $$speed of $$v = 11.1\ 	ext{m/s}. $$What is the magnitude of her centripetal acceleration?

Accepted Answer

$$a_c = \frac{v^2$}{r} = \frac{(11.1)^2$}{11} = 11.2\ 	ext{m/s}^2$$

Question 11

A plane leaves Chicago flying due east at $$500\ 	ext{mph}$$, while the wind blows south at $$100\ 	ext{mph}. $$What is the magnitude of the plane’s velocity relative to the ground?

Accepted Answer

$$\sqrt{500^2$ + 100^2$} = 509.9\ 	ext{mph}$$

Question 12

A skydiver falls at $$35\ 	ext{m/s}$$ vertically, with a wind blowing at $$7\ 	ext{m/s} to $$the west. What is the angle of his velocity with respect to vertical?

Accepted Answer

$$	heta = \arctan\left(\frac{7}{35}ight) = 11.3^\circ$$

Question 13

A projectile is launched at an angle $$	heta$$ with initial speed $$v_0. $$What is the time of flight if it lands at the same height from which it was launched?

Accepted Answer

$$t = \frac{2 v_0$ \sin 	heta}{g}$$

Question 14

A vector $$\vec{A}$$ has components $$A_x = 3$$ and $$A_y = 4. $$What is the magnitude of $$\vec{A}$$?

Accepted Answer

$$\sqrt{3^2$ + 4^2$} = 5$$

Question 15

A grasshopper jumps $$0.76\ 	ext{m} at $$the optimal angle for maximum range. What is the initial speed required?

Accepted Answer

$$v_0$$ = \sqrt{\frac{0.76 g}{\sin 2	heta}}$$ with $$	heta = 45^\circ$$, so v_0$ = \sqrt{0.76 	imes 9.8} = 2.73\ 	ext{m/s}$$

Question 16

A wind turbine blade rotates in a circle of radius $$40\ 	ext{m} at a $$speed of $$15\ 	ext{m/s}. $$What is the centripetal acceleration?

Accepted Answer

$$a_c = \frac{v^2$}{r} = \frac{225}{40} = 5.63\ 	ext{m/s}^2$$

Question 17

A bird flies at $$35\ 	ext{m/s}$$ relative to the wind, and the wind moves at $$23\ 	ext{m/s}$$ relative to the air. What is the bird’s speed relative to the air?

Accepted Answer

$$v_{ba} = v_{bw} - v_{aw} = 12\ 	ext{m/s}$$

Vectors and Motion in Two Dimensions: Study Guide

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