A water-skier is moving at a speed of 16.7 m/s. When she skis in the same direction as a traveling wave, she springs upward every 6.9 s because of the wave crests. When she skis in the direction opposite to the direction in which the wave moves, she springs upward every 4.2 s in response to the crests. The speed of the skier is greater than the speed of the wave.

Determine the speed of the wave in m/s and Determine the wavelength of the wave.

λ=(v-u) ∙t₁

λ=(v+u) ∙t₂
(v-u) ∙t₁=(v+u) ∙t₂
u=v(t₁-t₂)/(t₁+t₂) =16.7(6.9-4.2)/(6.9+4.2)=
=16.7∙2.7/11.1=4.06 ㎧
λ=(v+u) ∙t₂=(16.7+4.06) ∙4.2=87.2 m.

Why did the water-skier wear a life jacket? Because the waves gave her a high-five every few seconds!

Now, let's tackle your question. To find the speed of the wave, we can use the formula:

Wave speed = frequency × wavelength

We know the skier springs up every 6.9 s when skiing in the same direction as the wave, so the frequency is 1/6.9 Hz. Similarly, when skiing in the opposite direction, the frequency is 1/4.2 Hz.

Since the skier travels faster than the wave, we can assume that the skier's speed relative to the water is the wave speed.

Let's calculate the wave speed when the skier is skiing in the same direction:

Wave speed = frequency × period

Wave speed = (1/6.9 Hz) × (6.9 s) = 1 m/s

Now, let's calculate the wave speed when the skier is skiing in the opposite direction:

Wave speed = (1/4.2 Hz) × (4.2 s) ≈ 1 m/s

As you can see, the speed of the wave is the same in both cases, approximately 1 m/s.

Now, to determine the wavelength of the wave, we can use the formula:

Wave speed = frequency × wavelength

Wavelength = Wave speed / frequency

Wavelength = 1 m/s / (1/6.9 Hz) ≈ 6.9 m

Therefore, the wavelength of the wave is approximately 6.9 meters.

So there you have it! The speed of the wave is about 1 m/s, and the wavelength is approximately 6.9 meters.

To determine the speed of the wave, we can use the formula:

v = λ/T

where:
v is the velocity or speed of the wave,
λ is the wavelength of the wave, and
T is the period of the wave.

Given the skier's speed in the same and opposite directions, we have:

v_ski = 16.7 m/s
T_same = 6.9 s
T_opposite = 4.2 s

Let's first calculate the speed of the wave in the same direction as the skier:

v_same = λ / T_same

Rearranging the equation, we have:

λ = v_same * T_same

Substituting the given values, we get:

λ = (16.7 m/s) * (6.9 s)
λ = 115.23 m

Now, let's calculate the speed of the wave in the opposite direction as the skier:

v_opposite = λ / T_opposite

Rearranging the equation, we have:

λ = v_opposite * T_opposite

Substituting the given values, we get:

λ = (16.7 m/s) * (4.2 s)
λ = 70.14 m

Therefore, the speed of the wave is 115.23 m/s, and the wavelength of the wave is 70.14 m.

To solve this problem, we need to use the relationship between the speed of the wave, the frequency of the wave, and the wavelength of the wave.

Let's denote the speed of the wave as v, the frequency of the wave as f, and the wavelength as λ.

1. When the skier moves in the same direction as the wave:
The time period between successive wave crests is 6.9 s. Since the frequency (f) of a wave is the reciprocal of the time period (T), we can calculate the frequency:
f = 1 / T = 1 / 6.9 s = 0.145 Hz

The speed of the wave (v) can be calculated using the formula:
v = f * λ

Substituting the known values:
16.7 m/s = 0.145 Hz * λ

Solving for λ:
λ = 16.7 m/s / 0.145 Hz ≈ 115.17 m

Therefore, the wavelength of the wave is approximately 115.17 meters.

2. When the skier moves in the opposite direction of the wave:
The time period between successive wave crests is 4.2 s. Similarly, we can calculate the frequency:
f = 1 / T = 1 / 4.2 s = 0.2381 Hz

Using the same formula as before, we have:
16.7 m/s = 0.2381 Hz * λ

Solving for λ:
λ = 16.7 m/s / 0.2381 Hz ≈ 70.11 m

Therefore, the wavelength of the wave when the skier moves in the opposite direction is approximately 70.11 meters.

In summary:
- The speed of the wave is not given directly but can be calculated using the relationship v = f * λ. For the given information, the speed of the wave is not constant, but it changes depending on the motion of the skier.
- The wavelength of the wave is approximately 115.17 meters when the skier moves in the same direction as the wave, and approximately 70.11 meters when the skier moves in the opposite direction of the wave.