Question

Comprehensive Problems The speed of waves on a lake depends on frequency. For waves of frequency \(1.0 \mathrm{Hz}\), the wave speed is \(1.56 \mathrm{m} / \mathrm{s} ;\) for \(2.0-\mathrm{Hz}\) waves, the speed is \(0.78 \mathrm{m} / \mathrm{s} .\) The 2.0-Hz waves from a speedboat's wake reach you \(120 \mathrm{s}\) after the 1.0 -Hz waves generated by the same boat. How far away is the boat?

Short answer

The boat is 187.2 meters away.

Step-by-step solution

Understand the scenario

The problem involves waves of two different frequencies emanating from the same source and traveling across a lake. We need to calculate the distance of the source (the boat) from our position based on the time difference in the arrival of these waves.

Define the given parameters

We are given the wave speeds for the two frequencies: the speed for 1 Hz waves is 1.56 m/s and for 2 Hz waves is 0.78 m/s. The time difference between the arrivals of the two waves is 120 seconds.

Use the relationship between speed, distance, and time

The relationship is given by the equation \( \text{distance} = \text{speed} \times \text{time} \). We will use this formula to express the distance for both 1 Hz and 2 Hz waves, knowing that the distance traveled by both is the same.

Set up equations for both wave frequencies

Let \( d \) be the distance from the boat. For 1 Hz waves: \( d = 1.56 \times t_1 \). For 2 Hz waves: \( d = 0.78 \times t_2 \). Since the 2 Hz waves arrive 120 seconds after the 1 Hz waves, \( t_2 = t_1 + 120 \).

Solve equations simultaneously

Set the two distance expressions equal to each other: \[ 1.56t_1 = 0.78(t_1 + 120) \]Solving for \( t_1 \), expand the equation: \[ 1.56t_1 = 0.78t_1 + 93.6 \]. Rearrange to find \( t_1 \): \[ 1.56t_1 - 0.78t_1 = 93.6 \]. \[ 0.78t_1 = 93.6 \].\[ t_1 = \frac{93.6}{0.78} \approx 120 \text{ seconds} \].

Calculate the distance using one of the equations

Use \( t_1 = 120 \) seconds in the equation for 1 Hz waves: \[ d = 1.56 \times 120 = 187.2 \text{ meters} \].

Conclusion

Both waves travel the same distance, which we calculated to be 187.2 meters. This is the distance from the boat to your position.

Key concepts

Wave Frequency

Wave frequency is a core concept when studying how waves behave and move. The frequency of a wave is simply how often the waves pass a specific point in a given amount of time. The unit used to measure frequency is Hertz (Hz), which indicates the number of wave cycles that occur in one second. In the context of our problem, the two different wave frequencies (1 Hz and 2 Hz) determine how fast the waves travel across the lake:

• 1 Hz waves travel at a speed of 1.56 meters per second (m/s).
• 2 Hz waves travel more slowly at 0.78 m/s.

The difference in wave speed based on frequency is crucial for determining the time it takes for the waves to reach a particular point. Understanding frequency helps us connect this concept with how waves interact in various environments.

Distance Calculation

In order to understand how far the waves have traveled, knowing how to calculate distance becomes essential. Distance is a measure of how far one point is from another and can be calculated using the wave speeds and the time it takes for each wave to reach a given point.In our scenario, the distance wave travels can be calculated using:\[ \text{Distance} = \text{Speed} \times \text{Time} \]By using this formula with the information given:

• For 1 Hz waves: The distance traveled is calculated as \( 1.56 \times t_1 \).
• For 2 Hz waves: The distance is \( 0.78 \times t_2 \).

Both calculations rely on knowing the time it takes for the respective waves to travel. Setting these distance equations equal helps solve for the actual distance the waves have traveled, since both types of waves traverse the same distance.

Speed-Time-Distance Relationship

The interplay between speed, time, and distance is fundamentally important in wave problems. Understanding how these three variables relate can help you solve various problems involving motion.The basic relationship is expressed in the equation:\[ \text{Distance} = \text{Speed} \times \text{Time} \]This equation is applied by:

• Defining the wave speeds for different frequencies: Higher frequency waves can have different speeds due to environmental factors.
• Knowing the time it takes for each wave of known speed to cover a distance, which helps solve for unknowns like distance or time differences.

In the problem example, even though 2 Hz waves are slower, both 1 Hz and 2 Hz waves travel the same distance. Understanding how speed is inversely affected and solving the time equation helps in calculating the exact distance, as we equalize their travel scopes. Conclusions about real-world directions and behaviors of waves can be drawn by effectively applying this relationship through mathematical methods.