Q: A car drives up a hill inclined at $$20^\circ to $$the horizontal at a constant speed of $$15\ \mathrm{m/s}$$ for $$10\ \mathrm{s}. $$How much vertical height does the car gain?

$$h = 15 \times 10 \times \sin 20^\circ = 51.3\ \mathrm{m}$$

Q: Given a vector in the third quadrant with components $$(-4, -3)$$, what is the angle it makes with the positive x-axis?

$$\theta = 180^\circ + \tan$$^{-1}$$\left(\frac{-3}{-4}\right) = 216.87^\circ$$

Q: A projectile is launched with initial velocity $$v_0 at an $$angle $$\theta$$ above the horizontal. Which equation gives the time the projectile spends in the air (neglecting air resistance)?

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

Q: Which trigonometric function would you use to find the angle $$\theta of a $$vector given its components $$v_x$$ and $$v_y$$?

$$\tan \theta = \frac{v_y}{v_x}$$

Q: A vector $$\vec{B}$$ has components $$B_x = -5$$ and $$B_y = 12. $$What is the direction of $$\vec{B}$$ relative to the positive x-axis?

$$\theta = \tan$$^{-1}$$\left(\frac{12}{-5}\right) = 112.6^\circ$$

Question 1

A vector $$\vec{A}$$ has a magnitude of 10 units and makes an angle of $$30^\circ$$ with the positive x-axis. What are the x- and y-components of $$\vec{A}$$?

Accepted Answer

$$A_x = 10 \cos 30^\circ = 8.66$$, $$A_y = 10 \sin 30^\circ = 5$$

Question 2

Which mathematical relationship allows you to determine the magnitude of a vector $$\vec{v}$$ given its components $$v_x$$ and $$v_y$$?

Accepted Answer

Pythagorean’s theorem: $$v = \sqrt{v_x^2$ + v_y^2$}$$

Question 3

A car drives up a hill inclined at $$20^\circ to $$the horizontal at a constant speed of $$15\ \mathrm{m/s}$$ for $$10\ \mathrm{s}. $$How much vertical height does the car gain?

Accepted Answer

$$h = 15 	imes 10 	imes \sin 20^\circ = 51.3\ \mathrm{m}$$

Question 4

Given a vector in the third quadrant with components $$(-4, -3)$$, what is the angle it makes with the positive x-axis?

Accepted Answer

$$	heta = 180^\circ + 	an$$^{-1}$$\left(\frac{-3}{-4}ight) = 216.87^\circ$$

Question 5

A projectile is launched with initial velocity $$v_0 at an $$angle $$	heta$$ above the horizontal. Which equation gives the time the projectile spends in the air (neglecting air resistance)?

Accepted Answer

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

Question 6

Two projectiles are fired from a cannon with the same initial speed. Projectile A is launched at twice the angle of Projectile B. Which projectile stays in the air longer (neglecting air resistance)?

Accepted Answer

Projectile A

Question 7

A vector $$\vec{v}$$ has components $$v_x = -3$$ and $$v_y = 4. In $$which quadrant does this vector lie?

Accepted Answer

Quadrant II

Question 8

If $$\vec{A}$$ has components $$A_x = 4$$ and $$A_y = -3$$, what is the magnitude of $$\vec{A}$$?

Accepted Answer

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

Question 9

A ball is thrown horizontally from a cliff with speed $$v_0. $$Which of the following best describes the acceleration of the ball during its flight (neglecting air resistance)?

Accepted Answer

The acceleration is constant and downward, $$a = -g.$$

Question 10

A vector has a magnitude of $$7$$ and points in a direction $$225^\circ$$ from the positive x-axis. What are its x- and y-components?

Accepted Answer

$$A_x = 7 \cos 225^\circ = -4.95$$, $$A_y = 7 \sin 225^\circ = -4.95$$

Question 11

Which trigonometric function would you use to find the angle $$	heta of a $$vector given its components $$v_x$$ and $$v_y$$?

Accepted Answer

$$	an 	heta = \frac{v_y}{v_x}$$

Question 12

A projectile is launched at $$45^\circ$$ with initial speed $$v_0. $$What is the maximum height reached by the projectile?

Accepted Answer

$$h_{max} = \frac{v_0$^2 \sin^2$ 45^\circ}{2g}$$

Question 13

A vector $$\vec{B}$$ has components $$B_x = -5$$ and $$B_y = 12. $$What is the direction of $$\vec{B}$$ relative to the positive x-axis?

Accepted Answer

$$	heta = 	an$$^{-1}$$\left(\frac{12}{-5}ight) = 112.6^\circ$$

Question 14

A ball is thrown at an angle $$	heta$$ above the horizontal with initial speed $$v_0. $$Which component of velocity remains constant during the flight (neglecting air resistance)?

Accepted Answer

The horizontal component

Question 15

A vector in Quadrant IV has a positive x-component and a negative y-component. Which of the following could be its components?

Accepted Answer

$$(3, -4)$$

Question 16

A projectile is launched horizontally from a height $$h$$ with speed $$v_0. $$How long does it take to hit the ground?

Accepted Answer

$$t = \sqrt{\frac{2h}{g}}$$

Question 17

Which of the following best describes the relationship between the unit circle and trigonometric functions?

Accepted Answer

The x- and y-coordinates of a point on the unit circle correspond to $$\cos 	heta$$ and $$\sin 	heta$$, respectively.

Vector Components, Decomposition, and Projectile Motion – Physics with Calculus Study Guide

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