Analyze the data to identify the mathematical relationship between the amplitude and energy of a mechanical wave.

If mechanical wave [tex]$A$[/tex] has an amplitude of 1 cm and mechanical wave [tex]$B$[/tex] has an amplitude of 2 cm, what will be the relationship between the energy carried by the two waves?

\begin{tabular}{|l|l|}
\hline
Amplitude & Energy \\
\hline
1 unit & 2 units \\
\hline
2 units & 8 units \\
\hline
3 units & 18 units \\
\hline
4 units & 32 units \\
\hline
5 units & 50 units \\
\hline
\end{tabular}

A. The amount of energy in wave [tex]$B$[/tex] is four times the amount of energy in wave [tex]$A$[/tex].
B. The amount of energy in wave [tex]$A$[/tex] is four times the amount of energy in wave [tex]$B$[/tex].
C. The amount of energy in wave [tex]$B$[/tex] is half the amount of energy in wave [tex]$A$[/tex].
D. The amount of energy in wave [tex]$A$[/tex] is half the amount of energy in wave [tex]$B$[/tex].



Answer :

To identify the mathematical relationship between the amplitude and energy of a mechanical wave, let's carefully analyze the given data and determine the pattern that relates amplitude to energy.

Given data:
[tex]\[ \begin{array}{|l|l|} \hline \text{Amplitude (units)} & \text{Energy (units)} \\ \hline 1 & 2 \\ \hline 2 & 8 \\ \hline 3 & 18 \\ \hline 4 & 32 \\ \hline 5 & 50 \\ \hline \end{array} \][/tex]

Let's start by looking at the amplitudes and corresponding energies for two specific mechanical waves:
- Mechanical wave [tex]\(A\)[/tex] with amplitude [tex]\(1\)[/tex] unit has energy [tex]\(2\)[/tex] units.
- Mechanical wave [tex]\(B\)[/tex] with amplitude [tex]\(2\)[/tex] units has energy [tex]\(8\)[/tex] units.

To determine the relationship between the energy carried by these two waves, we must find the ratio of the energy of wave [tex]\(B\)[/tex] to the energy of wave [tex]\(A\)[/tex]:

1. Identify the energy of wave [tex]\(A\)[/tex]:
[tex]\[ E_A = 2\ \text{units} \][/tex]

2. Identify the energy of wave [tex]\(B\)[/tex]:
[tex]\[ E_B = 8\ \text{units} \][/tex]

3. Calculate the ratio of [tex]\(E_B\)[/tex] to [tex]\(E_A\)[/tex]:
[tex]\[ \frac{E_B}{E_A} = \frac{8}{2} = 4.0 \][/tex]

This ratio [tex]\(4.0\)[/tex] indicates that the energy in wave [tex]\(B\)[/tex] is four times the energy in wave [tex]\(A\)[/tex].

Therefore, the correct relationship is:
[tex]\[ \text{The amount of energy in wave } B \text{ is four times the amount of energy in wave } A. \][/tex]

Thus, the correct answer is:
[tex]\[ \mathbf{A. \text{The amount of energy in wave B is four times the amount of energy in wave A.}} \][/tex]