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.