Waves are coming towards your boat at a speed of 4 m/s, you are standing on the fish scale and notice that the reading on the scale changes up and down such that the lowest scale reading is ½ of the maximum scale reading. If the total vertical distance travelled by the boat as a result of being hit by the waves is 1.9 m, find the distance between two crests.

Question

Waves are coming towards your boat at a speed of 4 m/s, you are standing on the fish scale and notice that the reading on the scale changes up and down such that the lowest scale reading is ½ of the maximum scale reading. If the total vertical distance travelled by the boat as a result of being hit by the waves is 1.9 m, find the distance between two crests.

Answer 1

To find the distance between two crests, we need to first determine the amplitude of the waves. From the information given in the question, we know that the lowest scale reading is half of the maximum scale reading. This tells us that the amplitude of the waves is equal to the difference between the maximum and minimum scale readings divided by 2.

Let's denote the maximum scale reading as "A" and the minimum scale reading as "B". Given that B = (1/2) * A, we can substitute B in the formula for amplitude:

Amplitude (Amp) = (A - B) / 2
= (A - (1/2) * A) / 2
= (1/2) * A / 2
= A / 4

Now, we know that the total vertical distance traveled by the boat due to the waves is 1.9 m. This distance is equal to twice the amplitude (Amp) of the wave. So, we can set up the equation:

2 * Amp = 1.9

Substituting the value of Amp:

2 * (A / 4) = 1.9
A / 2 = 1.9
A = 3.8

Therefore, the maximum scale reading A is equal to 3.8 units.

Now, let's find the distance between two crests. The distance traveled by a wave in one complete oscillation (crest to crest or trough to trough) is called the wavelength (λ).

The wavelength is related to the speed of the wave (v) and its frequency (f) by the formula:

v = λ * f

Since the waves are coming towards the boat at a speed of 4 m/s, we can substitute the value of v:

4 = λ * f

We need to find the wavelength (λ). We also know that the frequency (f) of the wave is the number of crests passing a particular point in one second. Since the waves are periodic, we can assume that the time for one complete oscillation is 1/f.

Let's assume that the time for one complete oscillation is T. Therefore, 1/f = T.

The distance traveled by the boat in one complete oscillation is equal to 2 * Amp, which we found to be 3.8 units. Therefore:

2 * Amp = λ

Substituting the values:

2 * 3.8 = λ
λ = 7.6

So, the wavelength (distance between two crests) is 7.6 meters.