Q: A soccer ball is kicked from ground level with an initial speed of $$20\ \text{m/s} at an $$angle of $$30^\circ$$ above the horizontal. What is the initial vertical component of its velocity?

$$v_{0y} = 20 \sin(30^\circ) = 10\ \text{m/s}$$

Q: A projectile is launched horizontally from a $$45\ \text{m}$$ high cliff with a speed of $$12\ \text{m/s}. $$How long does it take to hit the ground? (Ignore air resistance.)

$$t = \sqrt{\dfrac{2 \times 45}{9.8}} \approx 3.03\ \text{s}$$

Q: A ball is thrown horizontally from a $$20\ \text{m}$$ high building. If it lands $$30\ \text{m}$$ away from the base, what was its initial speed? (Ignore air resistance.)

$$v_{0x} = \dfrac{30}{\sqrt{\dfrac{2 \times 20}{9.8}}}$$

Q: A projectile is launched from the ground at $$v_0 = 18$$\ \text{m/s}$$ at 45$$^\circ$$. What is the time to reach the maximum height?

$$t_{\text{peak}} = \dfrac{18 \sin(45^\circ)}{9.8}$$

Q: A projectile is launched at $$v_0 = 20$$\ \text{m/s}$$ at 30$$^\circ$$ above the horizontal. What is its maximum height?

$$\Delta y_{\text{max}} = \dfrac{(20 \sin(30^\$$circ))^2$$}{2 \times 9.8}$$

Q: A projectile is launched at $$v_0 = 22$$\ \text{m/s}$$ at 40$$^\circ$$ above the horizontal. What is the total time of flight if it lands at the same elevation?

$$t = \dfrac{2 \times 22 \sin(40^\circ)}{9.8}$$

Question 1

A soccer ball is kicked from ground level with an initial speed of $$20\ 	ext{m/s} at an $$angle of $$30^\circ$$ above the horizontal. What is the initial vertical component of its velocity?

Accepted Answer

$$v_{0y} = 20 \sin(30^\circ) = 10\ 	ext{m/s}$$

Question 2

A projectile is launched horizontally from a $$45\ 	ext{m}$$ high cliff with a speed of $$12\ 	ext{m/s}. $$How long does it take to hit the ground? (Ignore air resistance.)

Accepted Answer

$$t = \sqrt{\dfrac{2 	imes 45}{9.8}} \approx 3.03\ 	ext{s}$$

Question 3

Which statement best describes the independence of horizontal and vertical motions in projectile motion?

Accepted Answer

Horizontal and vertical motions are independent; the horizontal velocity does not affect the vertical fall.

Question 4

A ball is thrown horizontally from a $$20\ 	ext{m}$$ high building. If it lands $$30\ 	ext{m}$$ away from the base, what was its initial speed? (Ignore air resistance.)

Accepted Answer

$$v_{0x} = \dfrac{30}{\sqrt{\dfrac{2 	imes 20}{9.8}}}$$

Question 5

A projectile is launched at $$v_0 = 25$$\ 	ext{m/s}$$ at 60$$^\circ$$ above the horizontal. What is the horizontal range if it lands at the same elevation? (Use g = 9.8$$\ 	ext{m/s}^2$$.)

Accepted Answer

$$R = \dfrac{25^2$ \sin(120^\circ)}{9.8}$$

Question 6

At the highest point of a projectile's trajectory, which statement is true?

Accepted Answer

Vertical velocity is zero, but horizontal velocity remains constant.

Question 7

A projectile is launched from the ground at $$v_0 = 18$$\ 	ext{m/s}$$ at 45$$^\circ$$. What is the time to reach the maximum height?

Accepted Answer

$$t_{	ext{peak}} = \dfrac{18 \sin(45^\circ)}{9.8}$$

Question 8

A ball is dropped from rest and another is thrown horizontally from the same height. Which hits the ground first?

Accepted Answer

Both balls land at the same time.

Question 9

A projectile is launched at $$v_0 = 20$$\ 	ext{m/s}$$ at 30$$^\circ$$ above the horizontal. What is its maximum height?

Accepted Answer

$$\Delta y_{	ext{max}} = \dfrac{(20 \sin(30^\$$circ))^2$$}{2 	imes 9.8}$$

Question 10

A projectile is launched from the ground at $$v_0 = 15$$\ 	ext{m/s}$$ at 75$$^\circ$$. What is the horizontal component of its velocity?

Accepted Answer

$$v_{0x} = 15 \cos(75^\circ)$$

Question 11

A projectile is launched at $$v_0 = 22$$\ 	ext{m/s}$$ at 40$$^\circ$$ above the horizontal. What is the total time of flight if it lands at the same elevation?

Accepted Answer

$$t = \dfrac{2 	imes 22 \sin(40^\circ)}{9.8}$$

Question 12

A projectile is launched at $$v_0 = 10$$\ 	ext{m/s}$$ at 0$$^\circ$$ (horizontal) from a 5$$\ 	ext{m}$$ high platform. What is the horizontal distance traveled before hitting the ground?

Accepted Answer

$$x = 10 	imes \sqrt{\dfrac{2 	imes 5}{9.8}}$$

Question 13

If a projectile is launched at $$v_0 = 16$$\ 	ext{m/s}$$ at 60$$^\circ$$ and lands at the same elevation, what is its speed just before landing?

Accepted Answer

$$v = 16\ 	ext{m/s}$$

Question 14

A projectile is launched at $$v_0 = 12$$\ 	ext{m/s}$$ at 45$$^\circ$$ from ground level. What is the direction of its velocity just before landing?

Accepted Answer

$$	heta = 	an$$^{-1}$$\left(\dfrac{-v_{0y}}{v_{0x}}ight)$$

Question 15

Which launch angle gives the maximum range for a projectile landing at the same elevation as launch?

Accepted Answer

$$45^\circ$$

Question 16

A projectile is launched at $$v_0 = 14$$\ 	ext{m/s}$$ at 30$$^\circ$$ and lands at the same elevation. What is the range?

Accepted Answer

$$R = \dfrac{14^2$ \sin(60^\circ)}{9.8}$$

Question 17

A projectile is launched at $$v_0 = 20$$\ 	ext{m/s}$$ at 60$$^\circ$$ from ground level. What is the vertical velocity just before landing?

Accepted Answer

$$v_{y} = -20 \sin(60^\circ)$$

Question 18

A projectile is launched at $$v_0 = 10$$\ 	ext{m/s}$$ at 0$$^\circ$$ (horizontal) from a 4$$\ 	ext{m}$$ high platform. What is its vertical velocity just before hitting the ground?

Accepted Answer

$$v_{y} = -\sqrt{2 	imes 9.8 	imes 4}$$

Question 19

A projectile is launched at $$v_0 = 18$$\ 	ext{m/s}$$ at 30$$^\circ$$ from ground level. What is the total flight time?

Accepted Answer

$$t = \dfrac{2 	imes 18 \sin(30^\circ)}{9.8}$$

Two-Dimensional Kinematics and Projectile Motion: Study Notes

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