A person lying on an air mattress in the ocean rises and falls through one complete cycle every five seconds. The crests of the wave causing the motion are 20.0 m apart. Determine the frequency and the speed of the wave.

Options

A. 0.150 Hz, 15.0 m/s
B. 0.100 Hz, 8.00 m/s
C. 0.350 Hz, 2.00 m/s
D. 0.200 Hz, 4.00 m/s

Answer

To solve the problem, we need to determine the frequency and speed of the wave based on the information provided.

### Step 1: Determine the Frequency
The frequency (f) of a wave is defined as the number of cycles per second. It is the reciprocal of the time period (T), which is the time taken for one complete cycle.

Given:
- Time period (T) = 5 seconds

Using the formula:
\[ f = \frac{1}{T} \]

Substituting the value of T:
\[ f = \frac{1}{5} = 0.2 \text{ Hz} \]

### Step 2: Determine the Speed of the Wave
The speed (v) of a wave can be calculated using the formula:
\[ v = f \times \lambda \]
where:
- \( \lambda \) (wavelength) is the distance between two consecutive crests of the wave.

Given:
- Wavelength (\( \lambda \)) = 20.0 m

Now substituting the values we have:
\[ v = f \times \lambda = 0.2 \text{ Hz} \times 20.0 \text{ m} \]

Calculating the speed:
\[ v = 0.2 \times 20.0 = 4.0 \text{ m/s} \]

### Final Answers:
- Frequency (f) = 0.2 Hz
- Speed (v) = 4.0 m/s

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