To analyze the given data and identify the relationship between amplitude and energy, let's systematically break it down:
### Step-by-Step Analysis
1. List the Given Data Points
The table provided gives us the energy associated with different amplitudes:
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Amplitude} & \text{Energy} \\
\hline
1 \text{ unit} & 2 \text{ units} \\
\hline
2 \text{ units} & 8 \text{ units} \\
\hline
3 \text{ units} & 18 \text{ units} \\
\hline
4 \text{ units} & 32 \text{ units} \\
\hline
5 \text{ units} & 50 \text{ units} \\
\hline
\end{array}
\][/tex]
2. Identify Specific Amplitudes of Interest
We're particularly interested in analyzing the amplitudes of 1 unit and 2 units:
- Mechanical wave A has an amplitude of 1 unit.
- Mechanical wave B has an amplitude of 2 units.
3. Determine Corresponding Energy Values
From the table, we can extract the energy values directly for these amplitudes:
- The energy for amplitude 1 unit (wave A) is 2 units.
- The energy for amplitude 2 units (wave B) is 8 units.
4. Find the Relationship Between the Energies
To determine the relationship between the energies carried by the waves, we calculate the ratio of the energy of wave B to the energy of wave A. Specifically, we use the energy values found:
- Energy of wave A [tex]\( E_A = 2 \)[/tex] units.
- Energy of wave B [tex]\( E_B = 8 \)[/tex] units.
The relationship is:
[tex]\[
\text{Energy Relationship} = \frac{E_B}{E_A} = \frac{8 \text{ units}}{2 \text{ units}} = 4.0
\][/tex]
### Conclusion
The relationship between the energy carried by mechanical wave A (amplitude 1 unit) and mechanical wave B (amplitude 2 units) is that wave B carries 4 times the energy carried by wave A. This indicates a quadratic relationship between amplitude and energy, which is a common physical property: energy is often proportional to the square of the amplitude. Thus, when the amplitude doubles, the energy becomes four times greater.
