Question

analyze the data to identify the mathematical relationship between the amplitude and energy of a mechanical wave. if mechanical wave a has an amplitude of 4 cm and mechanical wave b has an amplitude of 5 cm, what will be the relationship between the energy carried by the two waves?

amplitude	energy
2 units	8 units
3 units	18 units
4 units	32 units
5 units	50 units

a. wave b has about 1.6 times more energy than wave a.
b. wave a has about 1.25 times more energy than wave b.
c. wave a has about 1.6 times more energy than wave b.
d. wave a has about 1.15 times more energy than wave b.

Explanation:

Step1: Identify energy - amplitude relationship

From the table, when amplitude $A = 1$, energy $E=2 = 2\times1^{2}$; when $A = 2$, $E = 8=2\times2^{2}$; when $A = 3$, $E = 18=2\times3^{2}$; when $A = 4$, $E = 32=2\times4^{2}$; when $A = 5$, $E = 50=2\times5^{2}$. The relationship between energy $E$ and amplitude $A$ is $E = 2A^{2}$.

Step2: Calculate energy of wave A

Wave A has amplitude $A_{A}=4$ cm. Using $E = 2A^{2}$, the energy of wave A, $E_{A}=2\times4^{2}=2\times16 = 32$ units.

Step3: Calculate energy of wave B

Wave B has amplitude $A_{B}=5$ cm. Using $E = 2A^{2}$, the energy of wave B, $E_{B}=2\times5^{2}=2\times25 = 50$ units.

Step4: Find the ratio of energies

The ratio $\frac{E_{B}}{E_{A}}=\frac{50}{32}=\frac{25}{16}=1.5625\approx1.6$.

Answer:

A. Wave B has about 1.6 times more energy than wave A.