swimming duck paddles the water with it's feet once every 1.6s, producing surface waves with this perios. The duck is moving with constant speed in a pond where the speed of the surface waves is 0.32m/s and the crests of the waves are 0.12m apart.
A) What is the duck's speed?
B)How far apart are the crests behind the duck
If you can only answer part of the question, that's okay. Thanks
It already says the crests of the waves are 0.12 m apart, but I suspect that may be for the forward direction. They will be farther apart behind the duck, because it moves away before the next paddle.
The duck creates a wave with each paddle. The wave moves 0.32 m/s * 1.6 s = 0.512 ahead with each paddle, but 0.12 m behind each crest there is a new wave, indicating that the duck moved 0.392 m in 1.6 s. The duck's speed is 0.392 m/1.6 s = 0.245 m/s. The duck is traveling at 76% of the wave speed, reducing the wave separation ahead of it.
Behind the duck, the distance between wave crests is the distance a wave travels in 1.6 s (0.512 m PLUS the distance that the duck has moved away in that time, 0.392 m. The total wave separation is 0.904 m.
To solve this problem, we can use the relationship between wave speed, frequency, and wavelength. The equation is:
wave speed = frequency x wavelength
Given information:
- Wave speed = 0.32 m/s
- Wavelength = 0.12 m
Let's solve each part of the question:
A) What is the duck's speed?
We know that the duck is paddling its feet once every 1.6 seconds, so the frequency can be calculated as the reciprocal of the period:
frequency = 1 / period
frequency = 1 / 1.6 s
frequency ≈ 0.625 Hz
Now, we can use the wave speed equation to find the duck's speed:
duck's speed = wave speed / frequency
duck's speed = 0.32 m/s / 0.625 Hz
duck's speed ≈ 0.512 m/s
Therefore, the duck's speed is approximately 0.512 m/s.
B) How far apart are the crests behind the duck?
Since the duck is moving forward, it creates a bow wave. The distance between the crests behind the duck can be calculated using the duck's speed and the wave speed:
distance = duck's speed x period
distance = 0.512 m/s x 1.6 s
distance ≈ 0.82 m
Therefore, the crests behind the duck are approximately 0.82 meters apart.
To find the duck's speed, we can analyze the relationship between the duck's paddle frequency and the speed of the surface waves.
A) The period of the surface waves is given as 1.6s, which represents the time taken for two consecutive crests to pass a fixed point. The speed of the surface waves is given as 0.32m/s.
The speed of a wave is determined by the equation:
Speed = Distance / Time
We can use this equation to find the distance between two consecutive crests:
Distance = Speed * Time
Distance = 0.32m/s * 1.6s
Distance = 0.512m
Since the distance between the crests is given as 0.12m, we can find the number of wave crests the duck produces in 1.6s:
Number of crests = Distance / Distance between crests
Number of crests = 0.512m / 0.12m
Number of crests = 4.27
Since the duck paddles once every 1.6s, it produces approximately 4 wave crests in that time period. To find the distance covered by the duck during this time, we multiply the number of crests by the distance between crests:
Distance covered by the duck = Number of crests * Distance between crests
Distance covered by the duck = 4.27 * 0.12m
Distance covered by the duck = 0.5124m
Therefore, the speed of the duck is approximately 0.5124 meters per 1.6 seconds, or 0.32m/s.
B) To find how far apart the crests are behind the duck, we need to consider the distance covered by the duck in 1.6 seconds:
Distance covered by the duck = Number of crests * Distance between crests
Distance covered by the duck = 4.27 * 0.12m
Distance covered by the duck = 0.5124m
Since the duck moves forward while producing the waves, the distance covered by the duck is the same as the distance between the crests behind it.
Therefore, the crests behind the duck are also 0.5124 meters apart.