Physics Quiz: Analyze Wave Amplitude And Energy

What this quiz covers

This quiz focuses on Analyze Wave Amplitude And Energy, giving you a quick way to practice the rules, question types, and explanations that matter most for Physics.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

Question 1

1. It increases by a factor of 2.
2. It increases by a factor of 4. (correct answer)
3. It increases by a factor of 8.
4. It decreases by a factor of 2.

Explanation: This question tests understanding of the relationship between wave amplitude and sound intensity. Wave energy is proportional to the amplitude squared: E ∝ A², which means if you double the amplitude (A → 2A), the energy increases by a factor of 4 (E → 4E), and if you triple the amplitude (A → 3A), the energy increases by a factor of 9 (E → 9E)—this squared relationship is fundamental to all types of waves including sound, water, seismic, and light waves. When amplitude doubles from A₁ to 2A₁, we calculate the intensity change using the squared relationship: I_new/I_original = (A_new/A_original)² = (2A₁/A₁)² = 2² = 4, so the intensity increases by a factor of 4 (quadruples), not by a factor of 2—this is why a sound that is twice as loud in amplitude requires 4 times as much energy to produce. Choice B is correct because it properly applies the squared relationship E ∝ A² to show energy factor is square of amplitude factor. Choice A incorrectly treats the relationship as linear (E ∝ A), claiming that doubling amplitude doubles energy, when actually the relationship is quadratic (E ∝ A²) so doubling amplitude quadruples energy—this is why the answer is a factor of 4, not 2. To solve amplitude-energy problems, remember the formula E₂/E₁ = (A₂/A₁)²: (1) if asked how energy changes when amplitude changes, square the amplitude factor (if A doubles, E increases by 2² = 4), (2) if asked how amplitude changes when energy changes, take the square root of the energy factor (if E increases 9-fold, A increases by √9 = 3), and (3) always check your answer makes sense—larger amplitude must mean more energy, and the relationship is stronger than linear (energy grows faster than amplitude). Physical intuition: the squared relationship makes sense because wave energy depends on both how far particles oscillate (amplitude) and how fast they oscillate back—both effects scale with amplitude, so total energy scales as amplitude squared; this is why increasing speaker volume slightly requires significantly more power, why tsunamis with modest height increase carry devastating energy, and why a magnitude 7 earthquake (10× amplitude of magnitude 6) releases roughly 32× more energy (actually $10^1$.5 in total energy released).

Question 2

1. $9$
2. $3$ (correct answer)
3. $\sqrt{9}/2$
4. $\tfrac{1}{3}$

Explanation: This question tests understanding of the relationship between wave amplitude and wave energy. Wave energy is proportional to the amplitude squared: E ∝ A², which means if you double the amplitude (A → 2A), the energy increases by a factor of 4 (E → 4E), and if you triple the amplitude (A → 3A), the energy increases by a factor of 9 (E → 9E)—this squared relationship is fundamental to all types of waves including sound, water, seismic, and light waves. If energy increases by a factor of 9, we solve for amplitude change: E₂/E₁ = (A₂/A₁)² = 9, taking square root of both sides gives A₂/A₁ = √9 = 3, so amplitude must increase by a factor of 3—for example, to increase energy 4-fold requires only doubling the amplitude because 2² = 4. Choice B is correct because it accurately uses square root relationship to find amplitude change from energy change. Choice A incorrectly treats the relationship as linear (E ∝ A), claiming that 9-fold energy requires 9-fold amplitude, when actually the relationship is quadratic (E ∝ A²) so 9-fold energy requires only 3-fold amplitude—this is why the answer is a factor of 3, not 9. To solve amplitude-energy problems, remember the formula E₂/E₁ = (A₂/A₁)²: (1) if asked how energy changes when amplitude changes, square the amplitude factor (if A doubles, E increases by 2² = 4), (2) if asked how amplitude changes when energy changes, take the square root of the energy factor (if E increases 9-fold, A increases by √9 = 3), and (3) always check your answer makes sense—larger amplitude must mean more energy, and the relationship is stronger than linear (energy grows faster than amplitude). Physical intuition: the squared relationship makes sense because wave energy depends on both how far particles oscillate (amplitude) and how fast they oscillate back—both effects scale with amplitude, so total energy scales as amplitude squared; this is why increasing speaker volume slightly requires significantly more power, why tsunamis with modest height increase carry devastating energy, and why a magnitude 7 earthquake (10× amplitude of magnitude 6) releases roughly 32× more energy (actually $10^1$.5 in total energy released).

Question 3

1. Energy is proportional to amplitude: $E \propto A$.
2. Energy is proportional to the square of amplitude: $E \propto A^2$. (correct answer)
3. Energy is inversely proportional to amplitude: $E \propto \tfrac{1}{A}$.
4. Energy is proportional to the cube of amplitude: $E \propto A^3$.

Explanation: This question tests understanding of the relationship between wave amplitude and wave energy. Wave energy is proportional to the amplitude squared: E ∝ A², which means if you double the amplitude (A → 2A), the energy increases by a factor of 4 (E → 4E), and if you triple the amplitude (A → 3A), the energy increases by a factor of 9 (E → 9E)—this squared relationship is fundamental to all types of waves including sound, water, seismic, and light waves. The mathematical relationship is E = kA² where k is a constant depending on the medium, frequency, and other wave properties—the key insight is the squared dependence, which appears in intensity of sound (I ∝ A²), power in water waves (P ∝ A²), and brightness of light (brightness ∝ A²), all reflecting the fundamental E ∝ A² relationship. Choice B is correct because it correctly identifies that energy is proportional to amplitude squared. Choice A incorrectly treats the relationship as linear (E ∝ A), claiming that doubling amplitude doubles energy, when actually the relationship is quadratic (E ∝ A²) so doubling amplitude quadruples energy—this is why the student's claim is wrong. To solve amplitude-energy problems, remember the formula E₂/E₁ = (A₂/A₁)²: (1) if asked how energy changes when amplitude changes, square the amplitude factor (if A doubles, E increases by 2² = 4), (2) if asked how amplitude changes when energy changes, take the square root of the energy factor (if E increases 9-fold, A increases by √9 = 3), and (3) always check your answer makes sense—larger amplitude must mean more energy, and the relationship is stronger than linear (energy grows faster than amplitude). Real-world applications of E ∝ A²: in sound, doubling the amplitude creates a sound that's 4 times more intense but only perceived as slightly louder due to logarithmic human hearing; in earthquakes, each step up in Richter scale represents 10× more amplitude but about 32× more energy (roughly $10^1$.5); in water waves, doubling wave height quadruples the energy, explaining why large tsunami waves are so devastating despite not looking enormously tall when viewed from space.

Question 4

1. Amplitude increased by a factor of 9
2. Amplitude increased by a factor of 4.5
3. Amplitude increased by a factor of 3 (correct answer)
4. Amplitude increased by a factor of $\sqrt{9}/2$

Explanation: This question tests understanding of the relationship between wave amplitude and wave energy. Wave energy is proportional to the amplitude squared: E ∝ A², which means if you double the amplitude (A → 2A), the energy increases by a factor of 4 (E → 4E), and if you triple the amplitude (A → 3A), the energy increases by a factor of 9 (E → 9E)—this squared relationship is fundamental to all types of waves including sound, water, seismic, and light waves. For energy increasing: If intensity (energy-related) increases by a factor of 9, we solve for amplitude change: I₂/I₁ = (A₂/A₁)² = 9, taking square root of both sides gives A₂/A₁ = √9 = 3, so amplitude must increase by a factor of 3—for example, to increase energy 4-fold requires only doubling the amplitude because 2² = 4. Choice C is correct because it accurately uses square root relationship to find amplitude change from energy change. Choice A incorrectly assumes energy is proportional to amplitude squared but then fails to take the square root, claiming a factor of 9 for amplitude when it's actually √9 = 3. To solve amplitude-energy problems, remember the formula E₂/E₁ = (A₂/A₁)²: (1) if asked how energy changes when amplitude changes, square the amplitude factor (if A doubles, E increases by 2² = 4), (2) if asked how amplitude changes when energy changes, take the square root of the energy factor (if E increases 9-fold, A increases by √9 = 3), and (3) always check your answer makes sense—larger amplitude must mean more energy, and the relationship is stronger than linear (energy grows faster than amplitude). Physical intuition: the squared relationship makes sense because wave energy depends on both how far particles oscillate (amplitude) and how fast they oscillate back—both effects scale with amplitude, so total energy scales as amplitude squared; this is why increasing speaker volume slightly requires significantly more power, why tsunamis with modest height increase carry devastating energy, and why a magnitude 7 earthquake (10× amplitude of magnitude 6) releases roughly 32× more energy (actually $10^1$.5 in energy scaling).

Question 5

1. $E_2/E_1 = 1/2$
2. $E_2/E_1 = 1/4$ (correct answer)
3. $E_2/E_1 = 2$
4. $E_2/E_1 = 4$

Explanation: This question tests understanding of the relationship between wave amplitude and wave energy. Wave energy is proportional to the amplitude squared: E ∝ A², which means if you double the amplitude (A → 2A), the energy increases by a factor of 4 (E → 4E), and if you triple the amplitude (A → 3A), the energy increases by a factor of 9 (E → 9E)—this squared relationship is fundamental to all types of waves including sound, water, seismic, and light waves. When amplitude halves from A₁ to A₂ = 0.5A₁, we calculate the energy change using the squared relationship: E₂/E₁ = (A₂/A₁)² = (0.5)² = 0.25 = 1/4, so the energy decreases to one-fourth, which is why reducing the pulse height by half reduces the energy carried to a quarter on the same rope. Choice B is correct because it properly applies the squared relationship E ∝ A² to show energy factor is square of amplitude factor. Choice A incorrectly treats the relationship as linear (E ∝ A), claiming that halving amplitude halves energy, when actually the relationship is quadratic (E ∝ A²) so halving amplitude quarters energy—this is why the answer is a factor of 1/4, not 1/2. To solve amplitude-energy problems, remember the formula E₂/E₁ = (A₂/A₁)²: (1) if asked how energy changes when amplitude changes, square the amplitude factor (if A doubles, E increases by 2² = 4), (2) if asked how amplitude changes when energy changes, take the square root of the energy factor (if E increases 9-fold, A increases by √9 = 3), and (3) always check your answer makes sense—larger amplitude must mean more energy, and the relationship is stronger than linear (energy grows faster than amplitude). Real-world applications of E ∝ A²: in sound, doubling the amplitude creates a sound that's 4 times more intense but only perceived as slightly louder due to logarithmic human hearing; in earthquakes, each step up in Richter scale represents 10× more amplitude but about 32× more energy (roughly $10^1$.5); in water waves, doubling wave height quadruples the energy, explaining why large tsunami waves are so devastating despite not looking enormously tall when viewed from space.

Question 6

1. Increases by a factor of 3
2. Increases by a factor of 9 (correct answer)
3. Decreases by a factor of 3
4. Increases by a factor of 6

Explanation: This question tests understanding of the relationship between wave amplitude and wave energy. Wave energy is proportional to the amplitude squared: E ∝ A², which means if you double the amplitude (A → 2A), the energy increases by a factor of 4 (E → 4E), and if you triple the amplitude (A → 3A), the energy increases by a factor of 9 (E → 9E)—this squared relationship is fundamental to all types of waves including sound, water, seismic, and light waves. When amplitude increases from 0.20 mm to 0.60 mm, we calculate the energy change using the squared relationship: E₂/E₁ = (A₂/A₁)² = (0.60/0.20)² = 3² = 9, so the energy increases by a factor of 9, which is why increasing the speaker's vibration extent by three times requires nine times more energy to produce the sound wave. Choice B is correct because it properly applies the squared relationship E ∝ A² to show the energy factor is the square of the amplitude factor. Choice A incorrectly treats the relationship as linear (E ∝ A), claiming that tripling amplitude increases energy by a factor of 3, when actually the relationship is quadratic (E ∝ A²) so tripling amplitude increases energy nine-fold—this is why the answer is a factor of 9, not 3. To solve amplitude-energy problems, remember the formula E₂/E₁ = (A₂/A₁)²: (1) if asked how energy changes when amplitude changes, square the amplitude factor (if A triples, E increases by 3² = 9), (2) if asked how amplitude changes when energy changes, take the square root of the energy factor (if E increases 9-fold, A increases by √9 = 3), and (3) always check your answer makes sense—larger amplitude must mean more energy, and the relationship is stronger than linear (energy grows faster than amplitude). Physical intuition: the squared relationship makes sense because wave energy depends on both how far particles oscillate (amplitude) and how fast they oscillate back—both effects scale with amplitude, so total energy scales as amplitude squared; this is why increasing speaker volume slightly requires significantly more power, why tsunamis with modest height increase carry devastating energy, and why a magnitude 7 earthquake (10× amplitude of magnitude 6) releases roughly 32× more energy (actually $10^1$.5 in energy scaling).

Question 7

1. $3$
2. $9$ (correct answer)
3. $\tfrac{1}{3}$
4. $\tfrac{1}{9}$

Explanation: This question tests understanding of the relationship between wave amplitude and wave energy in ocean waves. Wave energy is proportional to the amplitude squared: E ∝ A², which means if you double the amplitude (A → 2A), the energy increases by a factor of 4 (E → 4E), and if you triple the amplitude (A → 3A), the energy increases by a factor of 9 (E → 9E)—this squared relationship is fundamental to all types of waves including sound, water, seismic, and light waves. For comparing waves: Wave A with amplitude A_A = 1.5 m and Wave B with amplitude A_B = 0.5 m have energy ratio E_A/E_B = (A_A/A_B)² = (1.5/0.5)² = 3² = 9, meaning Wave A carries 9 times as much energy as Wave B despite having only three times the amplitude. Choice B is correct because it accurately calculates that tripling the amplitude increases energy 9-fold, properly applying the squared relationship E ∝ A². Choice A incorrectly treats the relationship as linear (E ∝ A), claiming the ratio is 3, when actually the relationship is quadratic (E ∝ A²) so the ratio is 9—this distinction is crucial because it determines that tripling amplitude increases energy 9-fold not 3-fold. To solve amplitude-energy problems, remember the formula E₂/E₁ = (A₂/A₁)²: if asked how energy changes when amplitude changes, square the amplitude factor (if A triples, E increases by 3² = 9), and always check your answer makes sense—larger amplitude must mean more energy, and the relationship is stronger than linear. Physical intuition: the squared relationship makes sense because wave energy depends on both how far particles oscillate (amplitude) and how fast they oscillate back—both effects scale with amplitude, so total energy scales as amplitude squared; this is why larger ocean waves can cause much more erosion and damage to coastlines.

Question 8

1. Energy is proportional to amplitude: $E \propto A$
2. Energy is proportional to amplitude squared: $E \propto A^2$ (correct answer)
3. Energy is inversely proportional to amplitude: $E \propto \tfrac{1}{A}$
4. Energy is proportional to amplitude cubed: $E \propto A^3$

Explanation: This question tests understanding of the fundamental relationship between wave amplitude and wave energy on a string. Wave energy is proportional to the amplitude squared: E ∝ A², which means if you double the amplitude (A → 2A), the energy increases by a factor of 4 (E → 4E), and if you triple the amplitude (A → 3A), the energy increases by a factor of 9 (E → 9E)—this squared relationship is fundamental to all types of waves including sound, water, seismic, and light waves. The mathematical relationship is E = kA² where k is a constant depending on the medium, frequency, and other wave properties—the key insight is the squared dependence, which appears in intensity of sound (I ∝ A²), power in water waves (P ∝ A²), and brightness of light (brightness ∝ A²), all reflecting the fundamental E ∝ A² relationship. Choice B is correct because it properly identifies that energy is proportional to amplitude squared (E ∝ A²). Choice A incorrectly treats the relationship as linear (E ∝ A), when actually the relationship is quadratic (E ∝ A²)—this distinction is crucial because it determines that doubling amplitude increases energy 4-fold not 2-fold. To solve amplitude-energy problems, remember the formula E₂/E₁ = (A₂/A₁)²: (1) if asked how energy changes when amplitude changes, square the amplitude factor (if A doubles, E increases by 2² = 4), (2) if asked how amplitude changes when energy changes, take the square root of the energy factor (if E increases 9-fold, A increases by √9 = 3), and (3) always check your answer makes sense—larger amplitude must mean more energy, and the relationship is stronger than linear. Physical intuition: the squared relationship makes sense because wave energy depends on both how far particles oscillate (amplitude) and how fast they oscillate back—both effects scale with amplitude, so total energy scales as amplitude squared; this is why a magnitude 7 earthquake (10× amplitude of magnitude 6) releases roughly 32× more energy (actually $~10^1$.5 in total energy released).

Question 9

1. Increases by a factor of 2
2. Increases by a factor of 4 (correct answer)
3. Decreases by a factor of 2
4. Increases by a factor of 8

Explanation: This question tests understanding of the relationship between wave amplitude and sound intensity, which is a measure of energy flux. Sound intensity is proportional to the amplitude squared: I ∝ A², meaning if you double the amplitude (A → 2A), the intensity increases by a factor of 4 (I → 4I), and this relationship holds for all waves where energy depends on the square of the displacement. When the amplitude increases from A₁ to 2A₁, we calculate the intensity change using the squared relationship: I_new/I_original = (2A₁/A₁)² = 2² = 4, so the intensity increases by a factor of 4, which explains why amplifying sound requires disproportionately more power. Choice B is correct because it properly applies the squared relationship I ∝ A² to show the intensity factor is the square of the amplitude factor. Choice A incorrectly treats the relationship as linear (I ∝ A), claiming that doubling amplitude doubles intensity, when actually the relationship is quadratic (I ∝ A²) so doubling amplitude quadruples intensity—this is why the answer is a factor of 4, not 2. To solve amplitude-energy problems, remember the formula I₂/I₁ = (A₂/A₁)²: if asked how intensity changes when amplitude changes, square the amplitude factor (if A doubles, I increases by 2² = 4), and always check your answer makes sense—larger amplitude must mean more intensity, and the relationship is stronger than linear. Physical intuition: the squared relationship makes sense because wave energy depends on both how far particles oscillate (amplitude) and how fast they oscillate back—both effects scale with amplitude, so total energy scales as amplitude squared; this is why increasing speaker volume slightly requires significantly more power.

Question 10

1. $3$
2. $9$ (correct answer)
3. $\tfrac{1}{9}$
4. $6$

Explanation: This question tests understanding of the relationship between wave amplitude and wave energy in water waves. Wave energy is proportional to the amplitude squared: E ∝ A², which means if you double the amplitude (A → 2A), the energy increases by a factor of 4 (E → 4E), and if you triple the amplitude (A → 3A), the energy increases by a factor of 9 (E → 9E)—this squared relationship is fundamental to all types of waves including sound, water, seismic, and light waves. For comparing waves: Wave 1 with amplitude A₁ = 0.20 m and Wave 2 with amplitude A₂ = 0.60 m have energy ratio E₂/E₁ = (A₂/A₁)² = (0.60/0.20)² = 3² = 9, meaning Wave 2 carries 9 times as much energy as Wave 1 despite having only three times the amplitude. Choice B is correct because it accurately calculates that tripling the amplitude increases energy 9-fold, properly applying the squared relationship E ∝ A². Choice A incorrectly treats the relationship as linear (E ∝ A), claiming the ratio is 3, when actually the relationship is quadratic (E ∝ A²) so the ratio is 9—this distinction is crucial because it determines that tripling amplitude increases energy 9-fold not 3-fold. To solve amplitude-energy problems, remember the formula E₂/E₁ = (A₂/A₁)²: if asked how energy changes when amplitude changes, square the amplitude factor (if A triples, E increases by 3² = 9), and always check your answer makes sense—larger amplitude must mean more energy, and the relationship is stronger than linear. Physical intuition: the squared relationship makes sense because wave energy depends on both how far particles oscillate (amplitude) and how fast they oscillate back—both effects scale with amplitude, so total energy scales as amplitude squared; this is why tsunamis with modest height increase carry devastating energy.

Question 11

1. It becomes $1/2$ as large
2. It becomes $1/4$ as large (correct answer)
3. It becomes 2 times as large
4. It becomes 4 times as large

Explanation: This question tests understanding of the relationship between wave amplitude and wave energy. Wave energy is proportional to the amplitude squared: E ∝ A², which means if you double the amplitude (A → 2A), the energy increases by a factor of 4 (E → 4E), and if you halve the amplitude (A → A/2), the energy decreases by a factor of 4 (E → E/4)—this squared relationship is fundamental to all types of waves including sound, water, seismic, and light waves. When amplitude is reduced to half its original value (A → A/2), we calculate the intensity change using the squared relationship: I₂/I₁ = (A₂/A₁)² = (A/2 / A)² = (1/2)² = 1/4, so the intensity becomes 1/4 as large—halving the amplitude reduces intensity to one-quarter of its original value. Choice B is correct because it properly applies the squared relationship I ∝ A² to show that halving amplitude reduces intensity to 1/4. Choice A incorrectly treats the relationship as linear (I ∝ A), claiming that halving amplitude halves intensity, when actually the relationship is quadratic (I ∝ A²) so halving amplitude reduces intensity to one-quarter—this is why the answer is 1/4, not 1/2. To solve amplitude-energy problems, remember the formula E₂/E₁ = (A₂/A₁)²: (1) if asked how energy changes when amplitude changes, square the amplitude factor (if A is halved, E becomes (1/2)² = 1/4), (2) if asked how amplitude changes when energy changes, take the square root of the energy factor (if E becomes 1/4, A becomes √(1/4) = 1/2), and (3) always check your answer makes sense—smaller amplitude must mean less energy, and the relationship is stronger than linear (energy decreases faster than amplitude). Physical intuition: the squared relationship makes sense because wave energy depends on both how far the electric field oscillates (amplitude) and how much energy is stored in the field—both effects scale with amplitude, so total intensity scales as amplitude squared; this is why dimming a light by reducing the electric field amplitude to half produces a light that appears much dimmer than half brightness, as it actually has only 1/4 the intensity.

Question 12

1. $5$
2. $10$
3. $25$ (correct answer)
4. $1/25$

Explanation: This question tests understanding of the relationship between wave amplitude and wave energy. Wave energy is proportional to the amplitude squared: E ∝ A², which means if you double the amplitude (A → 2A), the energy increases by a factor of 4 (E → 4E), and if you triple the amplitude (A → 3A), the energy increases by a factor of 9 (E → 9E)—this squared relationship is fundamental to all types of waves including sound, water, seismic, and light waves. For comparing waves: Wave X has amplitude 5 times that of Wave Y, so we calculate the energy ratio E_X/E_Y = (A_X/A_Y)² = 5² = 25—Wave X has 5 times the amplitude of Wave Y, so it carries 25 times as much energy despite having only 5 times the ground displacement. Choice C is correct because it properly applies the squared relationship E ∝ A² to show that having 5 times the amplitude means 25 times the energy. Choice A incorrectly treats the relationship as linear (E ∝ A), claiming that 5 times the amplitude means 5 times the energy, when actually the relationship is quadratic (E ∝ A²) so 5 times the amplitude means 25 times the energy—this is why the answer is a factor of 25, not 5. To solve amplitude-energy problems, remember the formula E₂/E₁ = (A₂/A₁)²: (1) if asked how energy changes when amplitude changes, square the amplitude factor (if A increases 5-fold, E increases by 5² = 25), (2) if asked how amplitude changes when energy changes, take the square root of the energy factor (if E increases 25-fold, A increases by √25 = 5), and (3) always check your answer makes sense—larger amplitude must mean more energy, and the relationship is stronger than linear (energy grows faster than amplitude). Real-world applications of E ∝ A²: in earthquakes, a seismic wave with 5 times the ground displacement carries 25 times more energy, explaining why seemingly modest increases in earthquake magnitude (which relates to amplitude) result in dramatically more destructive power—this is why a magnitude 7 earthquake (roughly 10× the amplitude of magnitude 6) releases about 32× more total energy.

Question 13

1. Energy is proportional to amplitude: $E \propto A$
2. Energy is proportional to amplitude squared: $E \propto A^2$ (correct answer)
3. Energy is inversely proportional to amplitude: $E \propto 1/A$
4. Energy is proportional to amplitude cubed: $E \propto A^3$

Explanation: This question tests understanding of the relationship between wave amplitude and wave energy. Wave energy is proportional to the amplitude squared: E ∝ A², which means if you double the amplitude (A → 2A), the energy increases by a factor of 4 (E → 4E), and if you triple the amplitude (A → 3A), the energy increases by a factor of 9 (E → 9E)—this squared relationship is fundamental to all types of waves including sound, water, seismic, and light waves. For identifying the relationship: The mathematical relationship is E = kA² where k is a constant depending on the medium, frequency, and other wave properties—the key insight is the squared dependence, which appears in intensity of sound (I ∝ A²), power in water waves (P ∝ A²), and brightness of light (brightness ∝ A²), all reflecting the fundamental E ∝ A² relationship stated in the problem. Choice B is correct because it correctly identifies that energy is proportional to amplitude squared, matching the given formula E = kA². Choice A incorrectly states a linear relationship E ∝ A, which would mean doubling amplitude doubles energy, when the correct relationship E ∝ A² means doubling amplitude quadruples energy—this distinction is crucial because it determines that increasing amplitude has a much stronger effect on energy than a simple proportional relationship would suggest. To solve amplitude-energy problems, remember the formula E₂/E₁ = (A₂/A₁)²: (1) if asked how energy changes when amplitude changes, square the amplitude factor (if A doubles, E increases by 2² = 4), (2) if asked how amplitude changes when energy changes, take the square root of the energy factor (if E increases 4-fold, A increases by √4 = 2), and (3) always check your answer makes sense—larger amplitude must mean more energy, and the relationship is stronger than linear (energy grows faster than amplitude). Physical intuition: the squared relationship makes sense because wave energy depends on both how far particles oscillate (amplitude) and how fast they oscillate back—both effects scale with amplitude, so total energy scales as amplitude squared; this universal E ∝ A² relationship applies to all wave types because it emerges from the fundamental physics of oscillating systems storing energy in both kinetic and potential forms.

Question 14

1. Increases by a factor of 1.5
2. Increases by a factor of 2.25 (correct answer)
3. Increases by a factor of 3.0
4. Decreases by a factor of 2.25

Explanation: This question tests understanding of the relationship between wave amplitude and wave energy. Wave energy is proportional to the amplitude squared: E ∝ A², which means if you double the amplitude (A → 2A), the energy increases by a factor of 4 (E → 4E), and if you increase amplitude by 1.5 times (A → 1.5A), the energy increases by a factor of 2.25 (E → 2.25E)—this squared relationship is fundamental to all types of waves including sound, water, seismic, and light waves. When amplitude becomes 1.5 times the original value, we calculate the energy change using the squared relationship: E₂/E₁ = (A₂/A₁)² = (1.5A/A)² = 1.5² = 2.25, so the energy increases by a factor of 2.25—increasing amplitude by 50% (factor of 1.5) increases energy by 125% (factor of 2.25). Choice B is correct because it properly applies the squared relationship E ∝ A² to show that increasing amplitude by a factor of 1.5 increases energy by 1.5² = 2.25. Choice A incorrectly treats the relationship as linear (E ∝ A), claiming that increasing amplitude by 1.5 times increases energy by 1.5 times, when actually the relationship is quadratic (E ∝ A²) so 1.5 times the amplitude means 2.25 times the energy—this is why the answer is a factor of 2.25, not 1.5. To solve amplitude-energy problems, remember the formula E₂/E₁ = (A₂/A₁)²: (1) if asked how energy changes when amplitude changes, square the amplitude factor (if A increases by 1.5, E increases by 1.5² = 2.25), (2) if asked how amplitude changes when energy changes, take the square root of the energy factor (if E increases 2.25-fold, A increases by √2.25 = 1.5), and (3) always check your answer makes sense—larger amplitude must mean more energy, and the relationship is stronger than linear (energy grows faster than amplitude). Real-world applications of E ∝ A²: in sound systems, increasing the speaker cone displacement by just 50% (1.5×) requires 2.25 times as much power—this explains why small increases in volume control can lead to significant increases in power consumption, and why audio amplifiers need substantial power reserves to handle even modest increases in output level.

Question 15

1. Increases by a factor of 3
2. Increases by a factor of 9 (correct answer)
3. Increases by a factor of 1.5
4. Decreases by a factor of 9

Explanation: This question tests understanding of the relationship between wave amplitude and wave energy. Wave energy is proportional to the amplitude squared: E ∝ A², which means if you double the amplitude (A → 2A), the energy increases by a factor of 4 (E → 4E), and if you triple the amplitude (A → 3A), the energy increases by a factor of 9 (E → 9E)—this squared relationship is fundamental to all types of waves including sound, water, seismic, and light waves. When amplitude increases from A₁ = 0.20 mm to A₂ = 0.60 mm, we calculate the energy change using the squared relationship: E₂/E₁ = (A₂/A₁)² = (0.60/0.20)² = 3² = 9, so the energy increases by a factor of 9—the amplitude tripled, so energy increased by 3² = 9. Choice B is correct because it properly applies the squared relationship E ∝ A² to show that tripling the amplitude increases energy by a factor of 9. Choice A incorrectly treats the relationship as linear (E ∝ A), claiming that tripling amplitude triples energy, when actually the relationship is quadratic (E ∝ A²) so tripling amplitude increases energy ninefold—this is why the answer is a factor of 9, not 3. To solve amplitude-energy problems, remember the formula E₂/E₁ = (A₂/A₁)²: (1) if asked how energy changes when amplitude changes, square the amplitude factor (if A triples, E increases by 3² = 9), (2) if asked how amplitude changes when energy changes, take the square root of the energy factor (if E increases 9-fold, A increases by √9 = 3), and (3) always check your answer makes sense—larger amplitude must mean more energy, and the relationship is stronger than linear (energy grows faster than amplitude). Real-world applications of E ∝ A²: in sound, tripling the amplitude creates a sound that's 9 times more intense but only perceived as moderately louder due to logarithmic human hearing; this explains why a speaker needs 9 times more power to triple the displacement of the speaker cone, making high-volume audio systems power-hungry.

Question 16

1. $3$
2. $9$ (correct answer)
3. $1/3$
4. $1/9$

Explanation: This question tests understanding of the relationship between wave amplitude and wave energy. Wave energy is proportional to the amplitude squared: E ∝ A², which means if you double the amplitude (A → 2A), the energy increases by a factor of 4 (E → 4E), and if you triple the amplitude (A → 3A), the energy increases by a factor of 9 (E → 9E)—this squared relationship is fundamental to all types of waves including sound, water, seismic, and light waves. For comparing waves: Wave A has amplitude A_A = 0.50 m and Wave B has amplitude A_B = 1.50 m, so we calculate the energy ratio E_B/E_A = (A_B/A_A)² = (1.50/0.50)² = 3² = 9—Wave B has three times the amplitude of Wave A, so it carries 9 times as much energy despite having only three times the height. Choice B is correct because it properly applies the squared relationship E ∝ A² to show that tripling amplitude increases energy 9-fold. Choice A incorrectly treats the relationship as linear (E ∝ A), claiming that tripling amplitude triples energy, when actually the relationship is quadratic (E ∝ A²) so tripling amplitude increases energy ninefold—this is why the answer is a factor of 9, not 3. To solve amplitude-energy problems, remember the formula E₂/E₁ = (A₂/A₁)²: (1) if asked how energy changes when amplitude changes, square the amplitude factor (if A triples, E increases by 3² = 9), (2) if asked how amplitude changes when energy changes, take the square root of the energy factor (if E increases 9-fold, A increases by √9 = 3), and (3) always check your answer makes sense—larger amplitude must mean more energy, and the relationship is stronger than linear (energy grows faster than amplitude). Real-world applications of E ∝ A²: in water waves, tripling wave height increases the energy ninefold, explaining why a 1.5-meter tsunami carries 9 times more energy than a 0.5-meter wave—this quadratic relationship is why even modest increases in tsunami height lead to dramatically more destructive power, as the force on structures and the ability to move debris scales with the wave's energy content.

Question 17

1. Energy is proportional to amplitude: $E \propto A$
2. Energy is proportional to the square of amplitude: $E \propto A^2$ (correct answer)
3. Energy is inversely proportional to amplitude: $E \propto 1/A$
4. Energy is proportional to the cube of amplitude: $E \propto A^3$

Explanation: This question tests understanding of the relationship between wave amplitude and wave energy. Wave energy is proportional to the amplitude squared: E ∝ A², which means if you double the amplitude (A → 2A), the energy increases by a factor of 4 (E → 4E), and if you triple the amplitude (A → 3A), the energy increases by a factor of 9 (E → 9E)—this squared relationship is fundamental to all types of waves including sound, water, seismic, and light waves. The mathematical relationship is E = kA² where k is a constant depending on the medium, frequency, and other wave properties—the key insight is the squared dependence, which appears in intensity of sound (I ∝ A²), power in water waves (P ∝ A²), and brightness of light (brightness ∝ A²), all reflecting the fundamental E ∝ A² relationship. Choice B is correct because it correctly identifies that energy is proportional to amplitude squared. Choice A incorrectly treats the relationship as linear (E ∝ A), claiming that doubling amplitude doubles energy, when actually the relationship is quadratic (E ∝ A²) so doubling amplitude quadruples energy—this distinction is crucial because it determines that tripling amplitude increases energy 9-fold (3² = 9) not 3-fold. To solve amplitude-energy problems, remember the formula E₂/E₁ = (A₂/A₁)²: (1) if asked how energy changes when amplitude changes, square the amplitude factor (if A doubles, E increases by 2² = 4), (2) if asked how amplitude changes when energy changes, take the square root of the energy factor (if E increases 9-fold, A increases by √9 = 3), and (3) always check your answer makes sense—larger amplitude must mean more energy, and the relationship is stronger than linear (energy grows faster than amplitude). Real-world applications of E ∝ A²: in sound, doubling the amplitude creates a sound that's 4 times more intense but only perceived as slightly louder due to logarithmic human hearing; in earthquakes, each step up in Richter scale represents 10× more amplitude but about 32× more energy (roughly $10^1$.5); in water waves, doubling wave height quadruples the energy, explaining why large tsunami waves are so devastating despite not looking enormously tall when viewed from space.

Question 18

1. 2
2. 4 (correct answer)
3. 8
4. 16

Explanation: This question tests understanding of the relationship between wave amplitude and wave energy. Wave energy is proportional to the amplitude squared: E ∝ A², which means if you double the amplitude (A → 2A), the energy increases by a factor of 4 (E → 4E), and if you triple the amplitude (A → 3A), the energy increases by a factor of 9 (E → 9E)—this squared relationship is fundamental to all types of waves including sound, water, seismic, and light waves. If intensity increases by a factor of 16, we solve for amplitude change: I₂/I₁ = (A₂/A₁)² = 16, taking square root of both sides gives A₂/A₁ = √16 = 4, so amplitude must increase by a factor of 4—to increase intensity 16-fold requires only quadrupling the amplitude because 4² = 16. Choice B is correct because it accurately uses square root relationship to find amplitude change from intensity change. Choice D incorrectly reverses the relationship, confusing the intensity factor (16) with the amplitude factor (4)—when intensity increases 16-fold, amplitude increases only 4-fold (not 16-fold) because we must take the square root: A₂/A₁ = √(I₂/I₁) = √16 = 4. To solve amplitude-energy problems, remember the formula E₂/E₁ = (A₂/A₁)²: (1) if asked how energy changes when amplitude changes, square the amplitude factor (if A doubles, E increases by 2² = 4), (2) if asked how amplitude changes when energy changes, take the square root of the energy factor (if E increases 9-fold, A increases by √9 = 3), and (3) always check your answer makes sense—larger amplitude must mean more energy, and the relationship is stronger than linear (energy grows faster than amplitude). Physical intuition: the squared relationship makes sense because wave energy depends on both how far particles oscillate (amplitude) and how fast they oscillate back—both effects scale with amplitude, so total energy scales as amplitude squared; this is why increasing speaker volume slightly requires significantly more power, why tsunamis with modest height increase carry devastating energy, and why a magnitude 7 earthquake (10× amplitude of magnitude 6) releases roughly 32× more energy (actually 10² in displacement but ~32× in total energy released).

Question 19

1. $1.5$
2. $2.25$ (correct answer)
3. $0.67$
4. $3.0$

Explanation: This question tests understanding of the relationship between wave amplitude and wave energy. Wave energy is proportional to the amplitude squared: E ∝ A², which means if you double the amplitude (A → 2A), the energy increases by a factor of 4 (E → 4E), and if you triple the amplitude (A → 3A), the energy increases by a factor of 9 (E → 9E)—this squared relationship is fundamental to all types of waves including sound, water, seismic, and light waves. Wave 1 with amplitude A₁ = 4 cm and Wave 2 with amplitude A₂ = 6 cm have energy ratio E₂/E₁ = (A₂/A₁)² = (6/4)² = (1.5)² = 2.25, meaning Wave 2 carries 2.25 times as much energy as Wave 1 despite having only 1.5 times the amplitude. Choice B is correct because it properly applies the squared relationship E ∝ A² to show energy factor is square of amplitude factor. Choice A incorrectly treats the relationship as linear (E ∝ A), calculating 6/4 = 1.5 instead of squaring it, when actually the relationship is quadratic (E ∝ A²) so the amplitude ratio must be squared—this is why the answer is 2.25, not 1.5. To solve amplitude-energy problems, remember the formula E₂/E₁ = (A₂/A₁)²: (1) if asked how energy changes when amplitude changes, square the amplitude factor (if A doubles, E increases by 2² = 4), (2) if asked how amplitude changes when energy changes, take the square root of the energy factor (if E increases 9-fold, A increases by √9 = 3), and (3) always check your answer makes sense—larger amplitude must mean more energy, and the relationship is stronger than linear (energy grows faster than amplitude). Physical intuition: the squared relationship makes sense because wave energy depends on both how far particles oscillate (amplitude) and how fast they oscillate back—both effects scale with amplitude, so total energy scales as amplitude squared; this is why increasing speaker volume slightly requires significantly more power, why tsunamis with modest height increase carry devastating energy, and why a magnitude 7 earthquake (10× amplitude of magnitude 6) releases roughly 32× more energy (actually 10² in displacement but ~32× in total energy released).

Question 20

1. 10
2. 20
3. 100 (correct answer)
4. 1/10

Explanation: This question tests understanding of the relationship between wave amplitude and wave energy. Wave energy is proportional to the amplitude squared: E ∝ A², which means if you double the amplitude (A → 2A), the energy increases by a factor of 4 (E → 4E), and if you increase amplitude 10-fold (A → 10A), the energy increases by a factor of 100 (E → 100E)—this squared relationship is fundamental to all types of waves including sound, water, seismic, and light waves. When amplitude increases from A₁ to A₂ = 10A₁, we calculate the energy change using the squared relationship: E₂/E₁ = (A₂/A₁)² = (10A₁/A₁)² = 10² = 100, so the energy increases by a factor of 100—this is why earthquakes with 10× larger ground motion are vastly more destructive, releasing 100× more energy. Choice C is correct because it properly applies the squared relationship E ∝ A² to show energy factor is square of amplitude factor. Choice A incorrectly treats the relationship as linear (E ∝ A), claiming that increasing amplitude 10-fold increases energy 10-fold, when actually the relationship is quadratic (E ∝ A²) so 10× amplitude means 100× energy—this is why the answer is a factor of 100, not 10. To solve amplitude-energy problems, remember the formula E₂/E₁ = (A₂/A₁)²: (1) if asked how energy changes when amplitude changes, square the amplitude factor (if A doubles, E increases by 2² = 4), (2) if asked how amplitude changes when energy changes, take the square root of the energy factor (if E increases 9-fold, A increases by √9 = 3), and (3) always check your answer makes sense—larger amplitude must mean more energy, and the relationship is stronger than linear (energy grows faster than amplitude). Physical intuition: the squared relationship makes sense because wave energy depends on both how far particles oscillate (amplitude) and how fast they oscillate back—both effects scale with amplitude, so total energy scales as amplitude squared; this is why increasing speaker volume slightly requires significantly more power, why tsunamis with modest height increase carry devastating energy, and why a magnitude 7 earthquake (10× amplitude of magnitude 6) releases roughly 32× more energy (actually 10² in displacement but ~32× in total energy released).